The Distribution of Patterns in Random Trees

Frédéric Chyzak*{}^{*}start_FLOATSUPERSCRIPT * end_FLOATSUPERSCRIPT, Michael Drmota**absent{}^{**}start_FLOATSUPERSCRIPT * * end_FLOATSUPERSCRIPT, Thomas Klausner**absent{}^{**}start_FLOATSUPERSCRIPT * * end_FLOATSUPERSCRIPT, and Gerard Kok***absent{}^{***}start_FLOATSUPERSCRIPT * * * end_FLOATSUPERSCRIPT
Abstract.

Let 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT denote the set of unrooted labeled trees of size n𝑛nitalic_n and let {\mathcal{M}}caligraphic_M be a particular (finite, unlabeled) tree. Assuming that every tree of 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is equally likely, it is shown that the limiting distribution as n𝑛nitalic_n goes to infinity of the number of occurrences of {\mathcal{M}}caligraphic_M as an induced subtree is asymptotically normal with mean value and variance asymptotically equivalent to μn𝜇𝑛\mu nitalic_μ italic_n and σ2nsuperscript𝜎2𝑛\sigma^{2}nitalic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n, respectively, where the constants μ>0𝜇0\mu>0italic_μ > 0 and σ0𝜎0\sigma\geq 0italic_σ ≥ 0 are computable.

*{}^{*}start_FLOATSUPERSCRIPT * end_FLOATSUPERSCRIPT INRIA-Rocquencourt, F-78153 Le Chesnay cedex, France, e-mail: frederic.chyzak@inria.fr
**absent{}^{**}start_FLOATSUPERSCRIPT * * end_FLOATSUPERSCRIPT Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/113, A-1040 Wien, Austria, e-mail: michael.drmota@tuwien.ac.at, klausner@dmg.tuwien.ac.at
***absent{}^{***}start_FLOATSUPERSCRIPT * * * end_FLOATSUPERSCRIPT Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, NL-2628 CD Delft, The Netherlands, e-mail: gkok@fsmat.at
This research was supported by the Austrian Science Foundation FWF, grants S8302 and S9604, and by the European Amadeus project.

1. Introduction

In this paper we consider unrooted labeled trees and analyse the number of occurrences of a tree pattern as an induced subtree of a random tree. It is well known that a typical tree in 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, the set of unrooted labeled trees of size n𝑛nitalic_n, has about μknsubscript𝜇𝑘𝑛\mu_{k}nitalic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_n nodes of degree k𝑘kitalic_k, where μk=1/e(k1)!subscript𝜇𝑘1𝑒𝑘1\mu_{k}=1/e(k-1)!italic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1 / italic_e ( italic_k - 1 ) !. Moreover, for any fixed k𝑘kitalic_k the total number of nodes of degree k𝑘kitalic_k over all trees in 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT satisfies a central limit theorem with mean and variance asymptotically equivalent to μknsubscript𝜇𝑘𝑛\mu_{k}nitalic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_n and σk2nsuperscriptsubscript𝜎𝑘2𝑛\sigma_{k}^{2}nitalic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n (for a specific constant σk>0subscript𝜎𝑘0\sigma_{k}>0italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT > 0). See [DG99], where Drmota and Gittenberger explored this phenomenon for unrooted labeled trees and other types of trees.

A node of degree k𝑘kitalic_k is an occurrence of what can be called a star with k𝑘kitalic_k edges. In this paper we continue this idea. We consider a pattern {\mathcal{M}}caligraphic_M, a given finite tree, and compute the limiting distribution of the number of occurrences of {\mathcal{M}}caligraphic_M in 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT as n𝑛n\to\inftyitalic_n → ∞. Note also that there can be overlaps of two or more copies of {\mathcal{M}}caligraphic_M, which we intend to count as separate occurrences.

Our main result in this paper is:

Theorem 1.

Let {\mathcal{M}}caligraphic_M be a given finite tree. Then the limiting distribution of the number of occurrences of {\mathcal{M}}caligraphic_M (as induced subtrees) in a tree of 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is asymptotically normal with mean and variance asymptotically equivalent to μn𝜇𝑛\mu nitalic_μ italic_n and σ2nsuperscript𝜎2𝑛\sigma^{2}nitalic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n, respectively, where μ>0𝜇0\mu>0italic_μ > 0 and σ20superscript𝜎20\sigma^{2}\geq 0italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 0 depend on the pattern {\mathcal{M}}caligraphic_M and can be computed explicitly and algorithmically and can be represented as polynomials (with rational coefficients) in 1/e1𝑒1/e1 / italic_e.

We consider here a random variable X𝑋Xitalic_X as Gaussian if its characteristic function is given by 𝐄eitX=eiμtσ2t2/2𝐄superscript𝑒𝑖𝑡𝑋superscript𝑒𝑖𝜇𝑡superscript𝜎2superscript𝑡22{\bf E}\,e^{itX}=e^{i\mu t-\sigma^{2}t^{2}/2}bold_E italic_e start_POSTSUPERSCRIPT italic_i italic_t italic_X end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT italic_i italic_μ italic_t - italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 end_POSTSUPERSCRIPT, that is, the case of zero variance σ2=0superscript𝜎20\sigma^{2}=0italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 is included here. For example, if {\mathcal{M}}caligraphic_M consists just of one edge (and two nodes), then the number of occurrences of {\mathcal{M}}caligraphic_M in 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is n1𝑛1n-1italic_n - 1 and thus constant. So in that particular case we have μ=1𝜇1\mu=1italic_μ = 1 and σ2=0superscript𝜎20\sigma^{2}=0italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0. Nevertheless we conjecture that σ2>0superscript𝜎20\sigma^{2}>0italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT > 0 in all other cases.

As already mentioned, the case of stars (or nodes of given degree) has been discussed in [DG99] for various classes of trees. Some previous work for unlabeled trees is due to Robinson and Schwenk [RS75]. Patterns in (rooted planar) trees have also been considered by Dershowitz and Zaks [DZ89] under the limitation that patterns start at the root. In a work on patterns in random binary search trees, Flajolet, Gourdon, and Martínez [FGM97] obtained a central limit theorem. Flajolet and Steyaert also analysed an algorithm for pattern matchings in trees [FS80a, FS80b, SF83]. Further Ruciǹski [Ruc88] established conditions for when the number of occurrences of a given subgraph in random graphs follows a normal distribution.

The plan of the paper is as follows. In Section 2 we give a short introduction to counting trees with generating functions, and also expand this to two variables for counting stars (nodes of specific degree k𝑘kitalic_k) in trees. In Section 3 we expand this framework to the counting of patterns in trees. The resulting asymptotics are presented in Section 4, concluding the proof of Theorem 1. Technical details for this as well as explicit algorithms can be found in the appendix. In fact, the algorithmic aspect is one of the driving forces of this paper.

2. Counting Trees and Counting Stars in Trees

In this section we introduce a three-step program to count the number of trees in 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}{}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT and in the same fashion the number of occurrences of nodes of degree k𝑘kitalic_k in 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}{}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. While redundant and probably heavy in this simplistic situation, this procedure was crucial to the derivation in [DG99] for counting stars and will generalise well to our setting of general tree patterns.

For this purpose we make use of the sets nsubscript𝑛{\mathcal{R}}_{n}caligraphic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT of rooted labeled trees of size n𝑛nitalic_n and 𝒫nsubscript𝒫𝑛{\mathcal{P}}_{n}caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT of planted labeled trees of size n𝑛nitalic_n. For rooted and unrooted trees, the size n𝑛nitalic_n counts the total number of nodes, whether internal or at the leaves. On the other hand, a planted tree is just a rooted tree where the root is adjoined an additional “phantom” node which does not contribute to the size of the tree, whereas the degree of the root is increased by one. As well, one can think of a planted tree as a rooted tree with an additional edge having no end vertex. The advantage of using planted trees, though it seems to add complexity, will be explained below. Obviously |𝒫n|=|n|subscript𝒫𝑛subscript𝑛|{\mathcal{P}}_{n}{}|=|{\mathcal{R}}_{n}{}|| caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = | caligraphic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | and |𝒯n|=|n|/nsubscript𝒯𝑛subscript𝑛𝑛|{\mathcal{T}}_{n}{}|=|{\mathcal{R}}_{n}{}|/n| caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = | caligraphic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | / italic_n. It is also well known that |n|=nn1subscript𝑛superscript𝑛𝑛1|{\mathcal{R}}_{n}{}|=n^{n-1}| caligraphic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = italic_n start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT and |𝒯n|=nn2subscript𝒯𝑛superscript𝑛𝑛2|{\mathcal{T}}_{n}{}|=n^{n-2}| caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = italic_n start_POSTSUPERSCRIPT italic_n - 2 end_POSTSUPERSCRIPT.

The three-step program is the following one: First, the generating function enumerating planted trees is determined, then it is used to count rooted trees by deriving their generating function, and finally the generating function counting unrooted trees is computed.

We define

p(x)=n=0|𝒫n|xnn!,r(x)=n=0|n|xnn!,t(x)=n=0|𝒯n|xnn!formulae-sequence𝑝𝑥superscriptsubscript𝑛0subscript𝒫𝑛superscript𝑥𝑛𝑛formulae-sequence𝑟𝑥superscriptsubscript𝑛0subscript𝑛superscript𝑥𝑛𝑛𝑡𝑥superscriptsubscript𝑛0subscript𝒯𝑛superscript𝑥𝑛𝑛p(x)=\sum_{n=0}^{\infty}|{\mathcal{P}}_{n}|\frac{x^{n}}{n!},\quad r(x)=\sum_{n=0}^{\infty}|{\mathcal{R}}_{n}|\frac{x^{n}}{n!},\quad t(x)=\sum_{n=0}^{\infty}|{\mathcal{T}}_{n}|\frac{x^{n}}{n!}italic_p ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT | caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG , italic_r ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT | caligraphic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG , italic_t ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT | caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG

and proceed in the following way:

  1. (1)

    Planted Rooted Trees: A planted tree is a planted root node with zero, one, two, \dots planted subtrees of any order. In terms of the generating function this yields

    p(x)=n=0xp(x)nn!=xep(x).𝑝𝑥superscriptsubscript𝑛0𝑥𝑝superscript𝑥𝑛𝑛𝑥superscript𝑒𝑝𝑥p(x)=\sum_{n=0}^{\infty}\frac{xp(x)^{n}}{n!}=xe^{p(x)}.italic_p ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_x italic_p ( italic_x ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG = italic_x italic_e start_POSTSUPERSCRIPT italic_p ( italic_x ) end_POSTSUPERSCRIPT .
  2. (2)

    Rooted Trees: For rooted trees we get the same (except for the phantom nodes which are not present here), just a root with zero, one, two, \dots planted subtrees of any order

    r(x)=n=0xp(x)nn!=xep(x)=p(x).𝑟𝑥superscriptsubscript𝑛0𝑥𝑝superscript𝑥𝑛𝑛𝑥superscript𝑒𝑝𝑥𝑝𝑥r(x)=\sum_{n=0}^{\infty}\frac{xp(x)^{n}}{n!}=xe^{p(x)}=p(x).italic_r ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_x italic_p ( italic_x ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG = italic_x italic_e start_POSTSUPERSCRIPT italic_p ( italic_x ) end_POSTSUPERSCRIPT = italic_p ( italic_x ) .
  3. (3)

    Unrooted Trees: Finally, we have |𝒯n|=|n|/nsubscript𝒯𝑛subscript𝑛𝑛|{\mathcal{T}}_{n}{}|=|{\mathcal{R}}_{n}{}|/n| caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = | caligraphic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | / italic_n, as already mentioned. However, we can also express t(x)𝑡𝑥t(x)italic_t ( italic_x ) by a relation which follows from a natural bijection between rooted trees on the one hand and unrooted trees and pairs of planted rooted trees (that are joined by identifying the additional edges at their planted roots and discarding the phantom nodes) on the other hand.111Consider the class of rooted (labeled) trees. If the root is labeled by 1111 then consider the tree as an unrooted tree. If the root is not labeled by 1111 then consider the first edge of the path between the root and 1111 and cut the tree into two planted rooted trees at this edge. This yields

    t(x)=r(x)12p(x)2.𝑡𝑥𝑟𝑥12𝑝superscript𝑥2t(x)=r(x)-\frac{1}{2}p(x)^{2}.italic_t ( italic_x ) = italic_r ( italic_x ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_p ( italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

The functional equation for p(x)𝑝𝑥p(x)italic_p ( italic_x ) can be either used to extract the explicit number |𝒫n|=nn1subscript𝒫𝑛superscript𝑛𝑛1|{\mathcal{P}}_{n}|=n^{n-1}| caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = italic_n start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT via Lagrange inversion or to obtain the radius of convergence and asymptotic expansions of the singular behaviour of this function. It is well known that x0=1/esubscript𝑥01𝑒x_{0}=1/eitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e is the common radius of convergence of p(x)𝑝𝑥p(x)italic_p ( italic_x ), r(x)𝑟𝑥r(x)italic_r ( italic_x ), and t(x)𝑡𝑥t(x)italic_t ( italic_x ), and that the singularity at x=x0𝑥subscript𝑥0x=x_{0}italic_x = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is of square-root type:

p(x)𝑝𝑥\displaystyle p(x)italic_p ( italic_x ) =r(x)=121ex+23(1ex)+,absent𝑟𝑥121𝑒𝑥231𝑒𝑥\displaystyle=r(x)=1-\sqrt{2}\sqrt{1-ex}+\frac{2}{3}(1-ex)+\cdots,= italic_r ( italic_x ) = 1 - square-root start_ARG 2 end_ARG square-root start_ARG 1 - italic_e italic_x end_ARG + divide start_ARG 2 end_ARG start_ARG 3 end_ARG ( 1 - italic_e italic_x ) + ⋯ ,
t(x)𝑡𝑥\displaystyle t(x)italic_t ( italic_x ) =12(1ex)+223(1ex)3/2+.absent121𝑒𝑥223superscript1𝑒𝑥32\displaystyle=\frac{1}{2}-(1-ex)+\frac{2\sqrt{2}}{3}(1-ex)^{3/2}+\cdots.= divide start_ARG 1 end_ARG start_ARG 2 end_ARG - ( 1 - italic_e italic_x ) + divide start_ARG 2 square-root start_ARG 2 end_ARG end_ARG start_ARG 3 end_ARG ( 1 - italic_e italic_x ) start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT + ⋯ .

This is reflected by the asymptotic expansions of the numbers

|𝒫n|subscript𝒫𝑛\displaystyle|{\mathcal{P}}_{n}{}|| caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | =|n|=nn1n!2πenn3/2,absentsubscript𝑛superscript𝑛𝑛1similar-to𝑛2𝜋superscript𝑒𝑛superscript𝑛32\displaystyle=|{\mathcal{R}}_{n}{}|=n^{n-1}\sim\frac{n!}{\sqrt{2\pi}}e^{n}n^{-3/2},= | caligraphic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = italic_n start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ∼ divide start_ARG italic_n ! end_ARG start_ARG square-root start_ARG 2 italic_π end_ARG end_ARG italic_e start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT - 3 / 2 end_POSTSUPERSCRIPT ,
|𝒯n|subscript𝒯𝑛\displaystyle|{\mathcal{T}}_{n}{}|| caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | =nn2n!2πenn5/2.absentsuperscript𝑛𝑛2similar-to𝑛2𝜋superscript𝑒𝑛superscript𝑛52\displaystyle=n^{n-2}\sim\frac{n!}{\sqrt{2\pi}}e^{n}n^{-5/2}.= italic_n start_POSTSUPERSCRIPT italic_n - 2 end_POSTSUPERSCRIPT ∼ divide start_ARG italic_n ! end_ARG start_ARG square-root start_ARG 2 italic_π end_ARG end_ARG italic_e start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT - 5 / 2 end_POSTSUPERSCRIPT .

In order to demonstrate the usefulness of the three-step procedure above we repeat the same steps for counting stars with k𝑘kitalic_k edges in trees, that is, the number of nodes of degree k𝑘kitalic_k, a given fixed positive number. Let pn,msubscript𝑝𝑛𝑚p_{n,m}italic_p start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT denote the number of planted trees of size n𝑛nitalic_n with exactly m𝑚mitalic_m nodes of degree k𝑘kitalic_k. Furthermore, let rn,msubscript𝑟𝑛𝑚r_{n,m}italic_r start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT and tn,msubscript𝑡𝑛𝑚t_{n,m}italic_t start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT be the corresponding numbers for rooted and unrooted trees and set

p(x,u)=n,m=0pn,mxnumn!,r(x,u)=n,m=0rn,mxnumn!,t(x,u)=n,m=0tn,mxnumn!.formulae-sequence𝑝𝑥𝑢superscriptsubscript𝑛𝑚0subscript𝑝𝑛𝑚superscript𝑥𝑛superscript𝑢𝑚𝑛formulae-sequence𝑟𝑥𝑢superscriptsubscript𝑛𝑚0subscript𝑟𝑛𝑚superscript𝑥𝑛superscript𝑢𝑚𝑛𝑡𝑥𝑢superscriptsubscript𝑛𝑚0subscript𝑡𝑛𝑚superscript𝑥𝑛superscript𝑢𝑚𝑛p(x,u)=\sum_{n,m=0}^{\infty}p_{n,m}\frac{x^{n}u^{m}}{n!},\quad r(x,u)=\sum_{n,m=0}^{\infty}r_{n,m}\frac{x^{n}u^{m}}{n!},\quad t(x,u)=\sum_{n,m=0}^{\infty}t_{n,m}\frac{x^{n}u^{m}}{n!}.italic_p ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT italic_n , italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG , italic_r ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT italic_n , italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG , italic_t ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT italic_n , italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG .

Then we have (compare with [DG99])

  1. (1)

    Planted Rooted Trees:

    p(x,u)=n=0nk1xp(x,u)nn!+xup(x,u)k1(k1)!=xep(x,u)+x(u1)p(x,u)k1(k1)!.𝑝𝑥𝑢superscriptsubscript𝑛0𝑛𝑘1𝑥𝑝superscript𝑥𝑢𝑛𝑛𝑥𝑢𝑝superscript𝑥𝑢𝑘1𝑘1𝑥superscript𝑒𝑝𝑥𝑢𝑥𝑢1𝑝superscript𝑥𝑢𝑘1𝑘1p(x,u)=\sum_{\begin{subarray}{c}n=0\\ n\neq k-1\end{subarray}}^{\infty}\frac{xp(x,u)^{n}}{n!}+\frac{xup(x,u)^{k-1}}{(k-1)!}=xe^{p(x,u)}+\frac{x(u-1)p(x,u)^{k-1}}{(k-1)!}.italic_p ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_n = 0 end_CELL end_ROW start_ROW start_CELL italic_n ≠ italic_k - 1 end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_x italic_p ( italic_x , italic_u ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG + divide start_ARG italic_x italic_u italic_p ( italic_x , italic_u ) start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_k - 1 ) ! end_ARG = italic_x italic_e start_POSTSUPERSCRIPT italic_p ( italic_x , italic_u ) end_POSTSUPERSCRIPT + divide start_ARG italic_x ( italic_u - 1 ) italic_p ( italic_x , italic_u ) start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_k - 1 ) ! end_ARG .
  2. (2)

    Rooted Trees:

    r(x,u)=n=0nkxp(x,u)nn!+xup(x,u)kk!=xep(x,u)+x(u1)p(x,u)kk!.𝑟𝑥𝑢superscriptsubscript𝑛0𝑛𝑘𝑥𝑝superscript𝑥𝑢𝑛𝑛𝑥𝑢𝑝superscript𝑥𝑢𝑘𝑘𝑥superscript𝑒𝑝𝑥𝑢𝑥𝑢1𝑝superscript𝑥𝑢𝑘𝑘r(x,u)=\sum_{\begin{subarray}{c}n=0\\ n\neq k\end{subarray}}^{\infty}\frac{xp(x,u)^{n}}{n!}+\frac{xup(x,u)^{k}}{k!}=xe^{p(x,u)}+\frac{x(u-1)p(x,u)^{k}}{k!}.italic_r ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_n = 0 end_CELL end_ROW start_ROW start_CELL italic_n ≠ italic_k end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_x italic_p ( italic_x , italic_u ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG + divide start_ARG italic_x italic_u italic_p ( italic_x , italic_u ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG start_ARG italic_k ! end_ARG = italic_x italic_e start_POSTSUPERSCRIPT italic_p ( italic_x , italic_u ) end_POSTSUPERSCRIPT + divide start_ARG italic_x ( italic_u - 1 ) italic_p ( italic_x , italic_u ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG start_ARG italic_k ! end_ARG .
  3. (3)

    Unrooted Trees: Similarly to the above we have tn,m=rn,m/nsubscript𝑡𝑛𝑚subscript𝑟𝑛𝑚𝑛t_{n,m}=r_{n,m}/nitalic_t start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT / italic_n which is sufficient for our purposes. However, as above, it is also possible to express t(x,u)𝑡𝑥𝑢t(x,u)italic_t ( italic_x , italic_u ) by

    t(x,u)=r(x,u)12p(x,u)2.𝑡𝑥𝑢𝑟𝑥𝑢12𝑝superscript𝑥𝑢2t(x,u)=r(x,u)-\frac{1}{2}p(x,u)^{2}.italic_t ( italic_x , italic_u ) = italic_r ( italic_x , italic_u ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_p ( italic_x , italic_u ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

Note that the use of the notion of planted trees is crucial in order to keep track of the nodes of degree k𝑘kitalic_k by means of the recursive structure of planted trees. In [DG99] this approach was used to show that the asymptotic distribution of the number of nodes of degree k𝑘kitalic_k in trees of size n𝑛nitalic_n is normal, with expectation and variance proportional to n𝑛nitalic_n.

3. Counting Patterns in Trees

We now generalize the counting procedure of Section 2 to more complicated patterns. For our purpose, a pattern is a given (finite unrooted unlabeled) tree {\mathcal{M}}caligraphic_M. To ease explanations, we will use as {\mathcal{M}}caligraphic_M the example graph in Figure 1.

Refer to caption
Figure 1. Example pattern

We say that a specific pattern {\mathcal{M}}caligraphic_M occurs in a tree T𝑇Titalic_T if {\mathcal{M}}caligraphic_M occurs in T𝑇Titalic_T as an induced subtree in the sense that the node degrees for the internal (filled) nodes in the pattern match the degrees of the corresponding nodes in T𝑇Titalic_T, while the external (empty) nodes match nodes of arbitrary degree.222More generally we could also consider pattern-matching problems for patterns in which some degrees of certain possibly external “filled” nodes must match exactly while the degrees of the other, possibly internal “empty” nodes might be different. But then the situation is more involved, see Section 5. Because the results for the patterns consisting of only one node or two nodes and one edge are trivial, we now concentrate on patterns with at least three nodes.

Our principal aim is to get relations for the generating functions which count the number of occurrences of a specific pattern {\mathcal{M}}caligraphic_M. Let pn,msubscript𝑝𝑛𝑚p_{n,m}italic_p start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT denote the number of planted rooted trees with n𝑛nitalic_n nodes and exactly m𝑚mitalic_m occurrences of the pattern {\mathcal{M}}{}caligraphic_M and let

p=p(x,u)=n,m=0pn,mxnumn!𝑝𝑝𝑥𝑢superscriptsubscript𝑛𝑚0subscript𝑝𝑛𝑚superscript𝑥𝑛superscript𝑢𝑚𝑛p=p(x,u)=\sum_{n,m=0}^{\infty}p_{n,m}\frac{x^{n}u^{m}}{n!}italic_p = italic_p ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT italic_n , italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG

be the corresponding generating function.

3.1. Generating Functions for Planted Rooted Trees

Proposition 1.

(Planted Rooted Trees) Let {\mathcal{M}}caligraphic_M be a pattern. Then there exists a certain number L+1𝐿1L+1italic_L + 1 of auxiliary functions aj(x,u)subscript𝑎𝑗𝑥𝑢a_{j}(x,u)italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , italic_u ) (0jL)0𝑗𝐿(0\leq j\leq L)( 0 ≤ italic_j ≤ italic_L ) with

p(x,u)=j=0Laj(x,u)𝑝𝑥𝑢superscriptsubscript𝑗0𝐿subscript𝑎𝑗𝑥𝑢p(x,u)=\sum_{j=0}^{L}a_{j}(x,u)italic_p ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , italic_u )

and polynomials Pj(y0,,yL,u)subscript𝑃𝑗subscript𝑦0normal-…subscript𝑦𝐿𝑢P_{j}(y_{0},\ldots,y_{L},u)italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) (1jL)1𝑗𝐿(1\leq j\leq L)( 1 ≤ italic_j ≤ italic_L ) with non-negative coefficients such that

(1) a0(x,u)subscript𝑎0𝑥𝑢\displaystyle a_{0}(x,u)italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) =xea0(x,u)++aL(x,u)xj=1LPj(a0(x,u),,aL(x,u),1)absent𝑥superscript𝑒subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢𝑥superscriptsubscript𝑗1𝐿subscript𝑃𝑗subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢1\displaystyle=xe^{a_{0}(x,u)+\cdots+a_{L}(x,u)}-x\sum_{j=1}^{L}P_{j}(a_{0}(x,u),\ldots,a_{L}(x,u),1)= italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) end_POSTSUPERSCRIPT - italic_x ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) , 1 )
a1(x,u)subscript𝑎1𝑥𝑢\displaystyle a_{1}(x,u)italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_u ) =xP1(a0(x,u),,aL(x,u),u)absent𝑥subscript𝑃1subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢𝑢\displaystyle=x\cdot P_{1}(a_{0}(x,u),\ldots,a_{L}(x,u),u)= italic_x ⋅ italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) , italic_u )
\displaystyle\ \vdots
aL(x,u)subscript𝑎𝐿𝑥𝑢\displaystyle a_{L}(x,u)italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) =xPL(a0(x,u),,aL(x,u),u).absent𝑥subscript𝑃𝐿subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢𝑢\displaystyle=x\cdot P_{L}(a_{0}(x,u),\ldots,a_{L}(x,u),u).= italic_x ⋅ italic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) , italic_u ) .

Furthermore,

j=1LPj(y0,,yL,1)cey0++yL,subscript𝑐superscriptsubscript𝑗1𝐿subscript𝑃𝑗subscript𝑦0subscript𝑦𝐿1superscript𝑒subscript𝑦0subscript𝑦𝐿\sum_{j=1}^{L}P_{j}(y_{0},\ldots,y_{L},1)\leq_{c}e^{y_{0}+\cdots+y_{L}},∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , 1 ) ≤ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ,

where fcgsubscript𝑐𝑓𝑔f\leq_{c}gitalic_f ≤ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_g means that all Taylor coefficients of the left-hand side are smaller than or equal to the corresponding coefficients of the right-hand side. Moreover, the dependency graph of this system is strongly connected.333The notion of dependency graph is explained in Appendix B and intuitively speaking, reflects the fact that no subsystem can be solved before the whole system.

The proof of this proposition is in fact the core of the paper. In order to make the arguments more transparent we will demonstrate them with the help of the example pattern in Figure 1. At each step of the proof we will also indicate how to make all constructions explicit so that it is possible to generate System (1) effectively.

Refer to caption
Figure 2. Planted pattern matching

In a first step we introduce the notion of a planted pattern. A planted pattern psubscript𝑝{\mathcal{M}}_{p}caligraphic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT is just a planted rooted tree where we again distinguish between internal (filled) and external (empty) nodes. It matches a planted rooted tree from 𝒯nsubscript𝒯𝑛{\mathcal{T}}_{n}caligraphic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT if psubscript𝑝{\mathcal{M}}_{p}caligraphic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT occurs as an induced subtree starting from the (planted) root, that is, the branch structure and node degrees of the filled nodes match. Two occurrences may overlap. For example, in Figure 2 the planted pattern psubscript𝑝{\mathcal{M}}_{p}caligraphic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT on the left matches the planted tree A𝐴Aitalic_A twice (following the left, resp. the right edge from the root), but B𝐵Bitalic_B not at all. Also remark that, notwithstanding the symmetry of C𝐶Citalic_C, the pattern psubscript𝑝{\mathcal{M}}_{p}caligraphic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT really matches C𝐶Citalic_C twice, as we are interested in matches in labeled trees.

Refer to caption
Figure 3. Planted patterns for the pattern in Figure 1

We now construct a planted pattern for each internal (filled) node of our pattern {\mathcal{M}}caligraphic_M which is adjacent to an external (empty) node. The internal (filled) node is considered as the planted root and one of the free attached leaves as the plant. In our example we obtain the two graphs in Figure 3.

The next step is to partition all planted trees according to their degree distribution up to some adequate level. To this end, let D𝐷Ditalic_D denote the set of out-degrees that occur in the planted patterns introduced above and hhitalic_h be the maximal height of these patterns. In our example we have D={2}𝐷2D=\{2\}italic_D = { 2 } and h=33h=3italic_h = 3. For obtaining a partition, we more precisely consider all trees of height less than or equal to hhitalic_h with out-degrees in D𝐷Ditalic_D. We distinguish two types of leaves in these trees, depending on the depth at which they appear: leaves in level hhitalic_h, denoted “\circ”, and leaves at levels less than hhitalic_h, denoted “\Box”. For our example we get 11 different trees a0,a1,,a10subscript𝑎0subscript𝑎1subscript𝑎10a_{0},a_{1},\ldots,a_{10}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT, depicted on Figure 4.

Refer to caption
Figure 4. Tree partition

These trees induce a natural partition of all planted trees for the following interpretation of the two types of leaves: We say that a tree T𝑇Titalic_T is contained in class444By abuse of notation the tree class corresponding to the finite tree ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is denoted by the same symbol ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT. ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT if it matches the finite tree (or pattern) ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT in such a way that a node of type \Box has degree not in D𝐷Ditalic_D, while a node of type \circ has any degree. For example, a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT corresponds to those planted trees where the out-degree of the root is not in D𝐷Ditalic_D.

It is easy to observe that these (obviously disjoint) classes of trees form a partition. Indeed, take any rooted tree. For any path from the root to a leaf, consider the first node with out-degree not in D𝐷Ditalic_D, and replace the whole subtree at it with \Box. Then replace any node at depth hhitalic_h with \circ. The tree obtained in this way is one in the list.

Furthermore, the classes above can be described recursively. To this end, it proves convenient to introduce a formal notation to describe operations between classes of trees: direct-sum\oplus denotes the disjoint union of classes; \setminus denotes set difference; recursive descriptions of tree classes are given in the form ai=xaj1e1ajesubscript𝑎𝑖𝑥superscriptsubscript𝑎subscript𝑗1subscript𝑒1superscriptsubscript𝑎subscript𝑗subscript𝑒a_{i}=xa_{j_{1}}^{e_{1}}\dotsm a_{j_{\ell}}^{e_{\ell}}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, to express that the class aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is constructed by attaching e1subscript𝑒1e_{1}italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT subtrees from the class aj1subscript𝑎subscript𝑗1a_{j_{1}}italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, e2subscript𝑒2e_{2}italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT subtrees from the class aj2subscript𝑎subscript𝑗2a_{j_{2}}italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, etc, to a root node that we denote x𝑥xitalic_x.

In our example we get the following relations:

a0subscript𝑎0\displaystyle a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =pi=110ai=xxi=010aixn=3(i=010ai)n,absent𝑝superscriptsubscriptdirect-sum𝑖110subscript𝑎𝑖direct-sum𝑥𝑥superscriptsubscriptdirect-sum𝑖010subscript𝑎𝑖𝑥superscriptsubscriptdirect-sum𝑛3superscriptsuperscriptsubscriptdirect-sum𝑖010subscript𝑎𝑖𝑛\displaystyle=p\setminus\bigoplus_{i=1}^{10}a_{i}=x\oplus x\bigoplus_{i=0}^{10}a_{i}\oplus x\bigoplus_{n=3}^{\infty}\biggl{(}\bigoplus_{i=0}^{10}a_{i}\biggr{)}^{n},= italic_p ∖ ⨁ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x ⊕ italic_x ⨁ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_x ⨁ start_POSTSUBSCRIPT italic_n = 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( ⨁ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ,
a1subscript𝑎1\displaystyle a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =xa02,absent𝑥superscriptsubscript𝑎02\displaystyle=xa_{0}^{2},= italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a2subscript𝑎2\displaystyle a_{2}italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =xa0a1,absent𝑥subscript𝑎0subscript𝑎1\displaystyle=xa_{0}a_{1},= italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ,
a3subscript𝑎3\displaystyle a_{3}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT =xa0(a2a3a4),absent𝑥subscript𝑎0direct-sumsubscript𝑎2subscript𝑎3subscript𝑎4\displaystyle=xa_{0}(a_{2}\oplus a_{3}\oplus a_{4}),= italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ,
a4subscript𝑎4\displaystyle a_{4}italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT =xa0(a5a6a7a8a9a10),absent𝑥subscript𝑎0direct-sumsubscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎8subscript𝑎9subscript𝑎10\displaystyle=xa_{0}(a_{5}\oplus a_{6}\oplus a_{7}\oplus a_{8}\oplus a_{9}\oplus a_{10}),= italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ) ,
a5subscript𝑎5\displaystyle a_{5}italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT =xa12,absent𝑥superscriptsubscript𝑎12\displaystyle=xa_{1}^{2},= italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a6subscript𝑎6\displaystyle a_{6}italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT =xa1(a2a3a4),absent𝑥subscript𝑎1direct-sumsubscript𝑎2subscript𝑎3subscript𝑎4\displaystyle=xa_{1}(a_{2}\oplus a_{3}\oplus a_{4}),= italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ,
a7subscript𝑎7\displaystyle a_{7}italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT =xa1(a5a6a7a8a9a10),absent𝑥subscript𝑎1direct-sumsubscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎8subscript𝑎9subscript𝑎10\displaystyle=xa_{1}(a_{5}\oplus a_{6}\oplus a_{7}\oplus a_{8}\oplus a_{9}\oplus a_{10}),= italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ) ,
a8subscript𝑎8\displaystyle a_{8}italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT =x(a2a3a4)2,absent𝑥superscriptdirect-sumsubscript𝑎2subscript𝑎3subscript𝑎42\displaystyle=x(a_{2}\oplus a_{3}\oplus a_{4})^{2},= italic_x ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a9subscript𝑎9\displaystyle a_{9}italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT =x(a2a3a4)(a5a6a7a8a9a10),absent𝑥direct-sumsubscript𝑎2subscript𝑎3subscript𝑎4direct-sumsubscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎8subscript𝑎9subscript𝑎10\displaystyle=x(a_{2}\oplus a_{3}\oplus a_{4})(a_{5}\oplus a_{6}\oplus a_{7}\oplus a_{8}\oplus a_{9}\oplus a_{10}),= italic_x ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ( italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ) ,
a10subscript𝑎10\displaystyle a_{10}italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT =x(a5a6a7a8a9a10)2.absent𝑥superscriptdirect-sumsubscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎8subscript𝑎9subscript𝑎102\displaystyle=x(a_{5}\oplus a_{6}\oplus a_{7}\oplus a_{8}\oplus a_{9}\oplus a_{10})^{2}.= italic_x ( italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

This is to be interpreted as follows. Trees in a1subscript𝑎1a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT consist of a (planted) root that is denoted by x𝑥xitalic_x that has out-degree 2222, and two children that are of out-degree distinct from 2222, that is, in a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Similarly, trees in a3subscript𝑎3a_{3}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT consist of a root x𝑥xitalic_x with out-degree 2222 and subject to the following additional constraints: one subtree at the root is exactly of type a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT; the other subtree, call it T𝑇Titalic_T, is of out-degree 2, either with both subtrees of degree other than 2222 (leading to T𝑇Titalic_T in a2subscript𝑎2a_{2}italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT), or with one subtree of degree 2222 and the other of degree other than 2222 (leading to T𝑇Titalic_T in a3subscript𝑎3a_{3}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT), or with both of its subtrees of degree 2222 (leading to T𝑇Titalic_T in class a4subscript𝑎4a_{4}italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT). Summarizing: a3=xa0(a2a3a4)subscript𝑎3𝑥subscript𝑎0direct-sumsubscript𝑎2subscript𝑎3subscript𝑎4a_{3}=xa_{0}(a_{2}\oplus a_{3}\oplus a_{4})italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ). Of course this can be also interpreted as a3=xa0a2xa0a3xa0a4subscript𝑎3direct-sum𝑥subscript𝑎0subscript𝑎2𝑥subscript𝑎0subscript𝑎3𝑥subscript𝑎0subscript𝑎4a_{3}=xa_{0}a_{2}\oplus xa_{0}a_{3}\oplus xa_{0}a_{4}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. Another more involved example corresponds to a8subscript𝑎8a_{8}italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT; here both subtrees are of the form a2a3a4direct-sumsubscript𝑎2subscript𝑎3subscript𝑎4a_{2}\oplus a_{3}\oplus a_{4}italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT.

To show that the recursive description can be obtained easily in general, consider a tree ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT obtained from some planted pattern psubscript𝑝{\mathcal{M}}_{p}caligraphic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT. Let s1subscript𝑠1s_{1}italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, …, sdsubscript𝑠𝑑s_{d}italic_s start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT denote its subtrees at the root. Then, in each sisubscript𝑠𝑖s_{i}italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, leaves of type \circ can appear only at level h11h-1italic_h - 1. Substitute for all such \circ either \Box or a node of out-degree chosen from D𝐷Ditalic_D and having \circ for all its subtrees. Do this substitution in all possible ways. The collection of trees obtained are some of the aksubscript𝑎𝑘a_{k}italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT’s, say ak1(j)subscript𝑎subscriptsuperscript𝑘𝑗1a_{k^{(j)}_{1}}italic_a start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, ak2(j)subscript𝑎subscriptsuperscript𝑘𝑗2a_{k^{(j)}_{2}}italic_a start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, etc. Thus, we obtain the recursive relation aj=x(ak1(1)ak2(1))(ak1(d)ak2(d))subscript𝑎𝑗𝑥direct-sumsubscript𝑎subscriptsuperscript𝑘11subscript𝑎subscriptsuperscript𝑘12direct-sumsubscript𝑎subscriptsuperscript𝑘𝑑1subscript𝑎subscriptsuperscript𝑘𝑑2a_{j}=x(a_{k^{(1)}_{1}}\oplus a_{k^{(1)}_{2}}\oplus\dotsb)\dotsm(a_{k^{(d)}_{1}}\oplus a_{k^{(d)}_{2}}\oplus\dotsb)italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊕ ⋯ ) ⋯ ( italic_a start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊕ ⋯ ) for ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT.

In general, we obtain a partition of L+1𝐿1L+1italic_L + 1 classes a0,,aLsubscript𝑎0subscript𝑎𝐿a_{0},\ldots,a_{L}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT and corresponding recursive descriptions, where each tree type ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT can be expressed as a disjoint union of tree classes of the kind

(2) xaj1ajr=xa0l0aLlL,𝑥subscript𝑎subscript𝑗1subscript𝑎subscript𝑗𝑟𝑥superscriptsubscript𝑎0subscript𝑙0superscriptsubscript𝑎𝐿subscript𝑙𝐿xa_{j_{1}}\dotsm a_{j_{r}}=xa_{0}^{l_{0}}\dotsm a_{L}^{l_{L}},italic_x italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ,

where r𝑟ritalic_r denotes the degree of the root of ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT and the non-negative integer lisubscript𝑙𝑖l_{i}italic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the number of repetitions of the tree type aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.

We proceed to show that this directly leads to a system of equations of the form (1), where each polynomial relation stems from a recursive equation between combinatorial classes.

Let ΛjsubscriptΛ𝑗\Lambda_{j}roman_Λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT be the set of tuples (l0,,lL)subscript𝑙0subscript𝑙𝐿(l_{0},\ldots,l_{L})( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) with the property that (l0,,lL)Λjsubscript𝑙0subscript𝑙𝐿subscriptΛ𝑗(l_{0},\ldots,l_{L})\in\Lambda_{j}( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) ∈ roman_Λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT if and only if the term of type (2) is involved in the recursive description of ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT (in expanded form). Further, let k=K(l0,,lL)𝑘𝐾subscript𝑙0subscript𝑙𝐿k=K(l_{0},\ldots,l_{L})italic_k = italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) denote the number of additional occurrences of the pattern {\mathcal{M}}{}caligraphic_M in (2) in the following sense: if b=xaj1ajr𝑏𝑥subscript𝑎subscript𝑗1subscript𝑎subscript𝑗𝑟b=xa_{j_{1}}\dotsm a_{j_{r}}italic_b = italic_x italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT and T𝑇Titalic_T is a (planted rooted) labeled tree of b𝑏bitalic_b with subtrees T1aj1subscript𝑇1subscript𝑎subscript𝑗1T_{1}\in a_{j_{1}}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, T2aj2subscript𝑇2subscript𝑎subscript𝑗2T_{2}\in a_{j_{2}}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, etc, and {\mathcal{M}}{}caligraphic_M occurs m1subscript𝑚1m_{1}italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT times in T1subscript𝑇1T_{1}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, m2subscript𝑚2m_{2}italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT times in T2subscript𝑇2T_{2}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, etc, then T𝑇Titalic_T contains {\mathcal{M}}{}caligraphic_M exactly m1+m2++md+ksubscript𝑚1subscript𝑚2subscript𝑚𝑑𝑘m_{1}+m_{2}+\cdots+m_{d}+kitalic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ⋯ + italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT + italic_k times. The number k𝑘kitalic_k corresponds to the number of occurrences of {\mathcal{M}}{}caligraphic_M in T𝑇Titalic_T in which the root of T𝑇Titalic_T occurs as internal node of the pattern. By construction of the classes aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT this number only depends on b𝑏bitalic_b and not on the particular tree Tb𝑇𝑏T\in bitalic_T ∈ italic_b. Let us clarify the calculation of k=K(l0,,lL)𝑘𝐾subscript𝑙0subscript𝑙𝐿k=K(l_{0},\dots,l_{L})italic_k = italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) with an example. Consider the class a9subscript𝑎9a_{9}italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT of the partition for the example pattern. Now, in order to determine the number of additional occurrences, we match the planted patterns of Figure 3 at the root of an arbitrary tree of class a9subscript𝑎9a_{9}italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT. The left planted pattern of Figure 3 matches three times, the right one matches once. Thus we find that in this case k=4𝑘4k=4italic_k = 4. For the other classes we find the following values of k=K(l0,,lL)𝑘𝐾subscript𝑙0subscript𝑙𝐿k=K(l_{0},\dots,l_{L})italic_k = italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ):

Terms of class a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT a1subscript𝑎1a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT a2subscript𝑎2a_{2}italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT a3subscript𝑎3a_{3}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT a4subscript𝑎4a_{4}italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT a5subscript𝑎5a_{5}italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT a6subscript𝑎6a_{6}italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT a7subscript𝑎7a_{7}italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT a8subscript𝑎8a_{8}italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT a9subscript𝑎9a_{9}italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT a10subscript𝑎10a_{10}italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT
Value of k𝑘kitalic_k 0 0 0 1 2 1 2 3 3 4 5

.

Now define series Pjsubscript𝑃𝑗P_{j}italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT by

Pj(y0,,yL,u)=(l0,,lL)Λj1l0!lL!y0l0yLlLuK(l0,,lL).subscript𝑃𝑗subscript𝑦0subscript𝑦𝐿𝑢subscriptsubscript𝑙0subscript𝑙𝐿subscriptΛ𝑗1subscript𝑙0subscript𝑙𝐿superscriptsubscript𝑦0subscript𝑙0superscriptsubscript𝑦𝐿subscript𝑙𝐿superscript𝑢𝐾subscript𝑙0subscript𝑙𝐿P_{j}(y_{0},\ldots,y_{L},u)=\sum_{(l_{0},\ldots,l_{L})\in\Lambda_{j}}\frac{1}{l_{0}!\cdots l_{L}!}y_{0}^{l_{0}}\cdots y_{L}^{l_{L}}u^{K(l_{0},\ldots,l_{L})}.italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) = ∑ start_POSTSUBSCRIPT ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) ∈ roman_Λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ! ⋯ italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ! end_ARG italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT .

These are in fact polynomials for 1jL1𝑗𝐿1\leq j\leq L1 ≤ italic_j ≤ italic_L by the finiteness of the corresponding ΛjsubscriptΛ𝑗\Lambda_{j}roman_Λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT. All matches of the planted patterns are handled in the Pjsubscript𝑃𝑗P_{j}italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, 1jL1𝑗𝐿1\leq j\leq L1 ≤ italic_j ≤ italic_L, thus

P0(y0,,yL,u)=ey0++yLj=1LPj(y0,,yL,1)subscript𝑃0subscript𝑦0subscript𝑦𝐿𝑢superscript𝑒subscript𝑦0subscript𝑦𝐿superscriptsubscript𝑗1𝐿subscript𝑃𝑗subscript𝑦0subscript𝑦𝐿1P_{0}(y_{0},\ldots,y_{L},u)=e^{y_{0}+\cdots+y_{L}}-\sum_{j=1}^{L}P_{j}(y_{0},\ldots,y_{L},1)italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) = italic_e start_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , 1 )

does not depend on u𝑢uitalic_u.

In our pattern we get for example for P8(y0,,y10,u)subscript𝑃8subscript𝑦0subscript𝑦10𝑢P_{8}(y_{0},\dots,y_{10},u)italic_P start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT , italic_u )

P8(y0,,y10,u)=12xy22u3+xy2y3u3+xy2y4u3+12xy32u3+xy3y4u3+12xy42u3=12x(y2+y3+y4)2u3.subscript𝑃8subscript𝑦0subscript𝑦10𝑢12𝑥superscriptsubscript𝑦22superscript𝑢3𝑥subscript𝑦2subscript𝑦3superscript𝑢3𝑥subscript𝑦2subscript𝑦4superscript𝑢312𝑥superscriptsubscript𝑦32superscript𝑢3𝑥subscript𝑦3subscript𝑦4superscript𝑢312𝑥superscriptsubscript𝑦42superscript𝑢312𝑥superscriptsubscript𝑦2subscript𝑦3subscript𝑦42superscript𝑢3P_{8}(y_{0},\dots,y_{10},u)=\frac{1}{2}xy_{2}^{2}u^{3}+xy_{2}y_{3}u^{3}+xy_{2}y_{4}u^{3}+\frac{1}{2}xy_{3}^{2}u^{3}+xy_{3}y_{4}u^{3}+\frac{1}{2}xy_{4}^{2}u^{3}=\frac{1}{2}x(y_{2}+y_{3}+y_{4})^{2}u^{3}.italic_P start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT , italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_x italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_x italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_x italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_y start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x ( italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT .

Finally, let aj;n,msubscript𝑎𝑗𝑛𝑚a_{j;n,m}italic_a start_POSTSUBSCRIPT italic_j ; italic_n , italic_m end_POSTSUBSCRIPT denote the number of planted rooted trees of type ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT with n𝑛nitalic_n nodes and m𝑚mitalic_m occurrences of the pattern {\mathcal{M}}{}caligraphic_M and set

aj(x,u)=n,m=0aj;n,mxnumn!.subscript𝑎𝑗𝑥𝑢superscriptsubscript𝑛𝑚0subscript𝑎𝑗𝑛𝑚superscript𝑥𝑛superscript𝑢𝑚𝑛a_{j}(x,u)=\sum_{n,m=0}^{\infty}a_{j;n,m}\frac{x^{n}u^{m}}{n!}.italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT italic_n , italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_j ; italic_n , italic_m end_POSTSUBSCRIPT divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG .

By this definition it is clear that

aj(x,u)=xPj(a0(x,u),,aL(x,u),u),subscript𝑎𝑗𝑥𝑢𝑥subscript𝑃𝑗subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢𝑢a_{j}(x,u)=x\cdot P_{j}\bigl{(}a_{0}(x,u),\ldots,a_{L}(x,u),u\bigr{)},italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_x ⋅ italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) , italic_u ) ,

because the size of labeled trees is counted by x𝑥xitalic_x (exponential generating function) and the occurrences of the patterns is additive and counted by u𝑢uitalic_u. Hence, we explicitly obtain the proposed structure of the system of functional equations (1).

For the example pattern we arrive at the following system of equations, where we denote the generating function of the class aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT by the same symbol aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT:

a0subscript𝑎0\displaystyle a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =a0(x,u)=pi=110ai=x+xi=010ai+xn=31n!(i=010ai)n,absentsubscript𝑎0𝑥𝑢𝑝superscriptsubscript𝑖110subscript𝑎𝑖𝑥𝑥superscriptsubscript𝑖010subscript𝑎𝑖𝑥superscriptsubscript𝑛31𝑛superscriptsuperscriptsubscript𝑖010subscript𝑎𝑖𝑛\displaystyle=a_{0}(x,u)=p-\sum_{i=1}^{10}a_{i}=x+x\sum_{i=0}^{10}a_{i}+x\sum_{n=3}^{\infty}\frac{1}{n!}\left(\sum_{i=0}^{10}a_{i}\right)^{n},= italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_p - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x + italic_x ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_x ∑ start_POSTSUBSCRIPT italic_n = 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_n ! end_ARG ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ,
a1subscript𝑎1\displaystyle a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =a1(x,u)=12xa02,absentsubscript𝑎1𝑥𝑢12𝑥superscriptsubscript𝑎02\displaystyle=a_{1}(x,u)=\frac{1}{2}xa_{0}^{2},= italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a2subscript𝑎2\displaystyle a_{2}italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =a2(x,u)=xa0a1,absentsubscript𝑎2𝑥𝑢𝑥subscript𝑎0subscript𝑎1\displaystyle=a_{2}(x,u)=xa_{0}a_{1},= italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ,
a3subscript𝑎3\displaystyle a_{3}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT =a3(x,u)=xa0(a2+a3+a4)u,absentsubscript𝑎3𝑥𝑢𝑥subscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎4𝑢\displaystyle=a_{3}(x,u)=xa_{0}(a_{2}+a_{3}+a_{4})u,= italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) italic_u ,
a4subscript𝑎4\displaystyle a_{4}italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT =a4(x,u)=xa0(a5+a6+a7+a8+a9+a10)u2,absentsubscript𝑎4𝑥𝑢𝑥subscript𝑎0subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎8subscript𝑎9subscript𝑎10superscript𝑢2\displaystyle=a_{4}(x,u)=xa_{0}(a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10})u^{2},= italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ) italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a5subscript𝑎5\displaystyle a_{5}italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT =a5(x,u)=12xa12u,absentsubscript𝑎5𝑥𝑢12𝑥superscriptsubscript𝑎12𝑢\displaystyle=a_{5}(x,u)=\frac{1}{2}xa_{1}^{2}u,= italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x , italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u ,
a6subscript𝑎6\displaystyle a_{6}italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT =a6(x,u)=xa1(a2+a3+a4)u2,absentsubscript𝑎6𝑥𝑢𝑥subscript𝑎1subscript𝑎2subscript𝑎3subscript𝑎4superscript𝑢2\displaystyle=a_{6}(x,u)=xa_{1}(a_{2}+a_{3}+a_{4})u^{2},= italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a7subscript𝑎7\displaystyle a_{7}italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT =a7(x,u)=xa1(a5+a6+a7+a8+a9+a10)u3,absentsubscript𝑎7𝑥𝑢𝑥subscript𝑎1subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎8subscript𝑎9subscript𝑎10superscript𝑢3\displaystyle=a_{7}(x,u)=xa_{1}(a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10})u^{3},= italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ) italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ,
a8subscript𝑎8\displaystyle a_{8}italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT =a8(x,u)=12x(a2+a3+a4)2u3,absentsubscript𝑎8𝑥𝑢12𝑥superscriptsubscript𝑎2subscript𝑎3subscript𝑎42superscript𝑢3\displaystyle=a_{8}(x,u)=\frac{1}{2}x(a_{2}+a_{3}+a_{4})^{2}u^{3},= italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ( italic_x , italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ,
a9subscript𝑎9\displaystyle a_{9}italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT =a9(x,u)=x(a2+a3+a4)(a5+a6+a7+a8+a9+a10)u4,absentsubscript𝑎9𝑥𝑢𝑥subscript𝑎2subscript𝑎3subscript𝑎4subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎8subscript𝑎9subscript𝑎10superscript𝑢4\displaystyle=a_{9}(x,u)=x(a_{2}+a_{3}+a_{4})(a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10})u^{4},= italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_x ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ( italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ) italic_u start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ,
a10subscript𝑎10\displaystyle a_{10}italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT =a10(x,u)=12x(a5+a6+a7+a8+a9+a10)2u5.absentsubscript𝑎10𝑥𝑢12𝑥superscriptsubscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎8subscript𝑎9subscript𝑎102superscript𝑢5\displaystyle=a_{10}(x,u)=\frac{1}{2}x(a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10})^{2}u^{5}.= italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_x , italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x ( italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT .

In order to complete the proof of Proposition 1 we just have to show that the dependency graph is strongly connected. By construction, a0=a0(x,u)subscript𝑎0subscript𝑎0𝑥𝑢a_{0}=a_{0}(x,u)italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) depends on all functions ai=ai(x,u)subscript𝑎𝑖subscript𝑎𝑖𝑥𝑢a_{i}=a_{i}(x,u)italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_u ). Thus, it is sufficient to prove that every aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (1iL1𝑖𝐿1\leq i\leq L1 ≤ italic_i ≤ italic_L) also depends on a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. For this purpose consider the subtree of {\mathcal{M}}caligraphic_M that was labeled by aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and consider a path from its root to an empty node. Each edge of this path corresponds to another subtree of {\mathcal{M}}caligraphic_M, say ai2subscript𝑎subscript𝑖2a_{i_{2}}italic_a start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, ai3,,airsubscript𝑎subscript𝑖3subscript𝑎subscript𝑖𝑟a_{i_{3}},\ldots,a_{i_{r}}italic_a start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Then, by construction of the system of functional equations above, aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT depends on ai2subscript𝑎subscript𝑖2a_{i_{2}}italic_a start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, ai2subscript𝑎subscript𝑖2a_{i_{2}}italic_a start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT depends on ai3subscript𝑎subscript𝑖3a_{i_{3}}italic_a start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT etc. Finally the root of airsubscript𝑎subscript𝑖𝑟a_{i_{r}}italic_a start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT is adjacent to an empty node and thus (the corresponding generating function) depends on a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. This completes the proof of Proposition 1.


Note that we obtain a relatively more compact form of this system by introducing

(3) b0subscript𝑏0\displaystyle b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =b0(x,u)=a0(x,u),absentsubscript𝑏0𝑥𝑢subscript𝑎0𝑥𝑢\displaystyle=b_{0}(x,u)=a_{0}(x,u),= italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) ,
b1subscript𝑏1\displaystyle b_{1}italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =b1(x,u)=a1(x,u),absentsubscript𝑏1𝑥𝑢subscript𝑎1𝑥𝑢\displaystyle=b_{1}(x,u)=a_{1}(x,u),= italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_u ) ,
b2subscript𝑏2\displaystyle b_{2}italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =b2(x,u)=a2(x,u)+a3(x,u)+a4(x,u)absentsubscript𝑏2𝑥𝑢subscript𝑎2𝑥𝑢subscript𝑎3𝑥𝑢subscript𝑎4𝑥𝑢\displaystyle=b_{2}(x,u)=a_{2}(x,u)+a_{3}(x,u)+a_{4}(x,u)= italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_u ) + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_x , italic_u ) + italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_x , italic_u )
b3subscript𝑏3\displaystyle b_{3}italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT =b3(x,u)=a5(x,u)+a6(x,u)+a7(x,u)+a8(x,u)+a9(x,u)+a10(x,u),absentsubscript𝑏3𝑥𝑢subscript𝑎5𝑥𝑢subscript𝑎6𝑥𝑢subscript𝑎7𝑥𝑢subscript𝑎8𝑥𝑢subscript𝑎9𝑥𝑢subscript𝑎10𝑥𝑢\displaystyle=b_{3}(x,u)=a_{5}(x,u)+a_{6}(x,u)+a_{7}(x,u)+a_{8}(x,u)+a_{9}(x,u)+a_{10}(x,u),= italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x , italic_u ) + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x , italic_u ) + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ( italic_x , italic_u ) + italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ( italic_x , italic_u ) + italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ( italic_x , italic_u ) + italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_x , italic_u ) ,

together with the recursive relations

b0subscript𝑏0\displaystyle b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =xeb0+b1+b2+b312x(b0+b1+b2+b3)2,absent𝑥superscript𝑒subscript𝑏0subscript𝑏1subscript𝑏2subscript𝑏312𝑥superscriptsubscript𝑏0subscript𝑏1subscript𝑏2subscript𝑏32\displaystyle=xe^{b_{0}+b_{1}+b_{2}+b_{3}}-\frac{1}{2}x(b_{0}+b_{1}+b_{2}+b_{3})^{2},= italic_x italic_e start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x ( italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
b1subscript𝑏1\displaystyle b_{1}italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =12xb02,absent12𝑥superscriptsubscript𝑏02\displaystyle=\frac{1}{2}xb_{0}^{2},= divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
b2subscript𝑏2\displaystyle b_{2}italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =xb0b1+xb0b2u+xb0b3u2,absent𝑥subscript𝑏0subscript𝑏1𝑥subscript𝑏0subscript𝑏2𝑢𝑥subscript𝑏0subscript𝑏3superscript𝑢2\displaystyle=xb_{0}b_{1}+xb_{0}b_{2}u+xb_{0}b_{3}u^{2},= italic_x italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_x italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_u + italic_x italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
b3subscript𝑏3\displaystyle b_{3}italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT =12xb12u+xb1b2u2+xb1b3u3+12xb22u3++xb2b3u4+12xb32u5.\displaystyle=\frac{1}{2}xb_{1}^{2}u+xb_{1}b_{2}u^{2}+xb_{1}b_{3}u^{3}+\frac{1}{2}xb_{2}^{2}u^{3}++xb_{2}b_{3}u^{4}+\frac{1}{2}xb_{3}^{2}u^{5}.= divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u + italic_x italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_x italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + + italic_x italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT .

The combinatorial classes corresponding to the bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (which we will also denote by bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT) have the interpretation shown in Figure 5. We could have obtained the classes bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT directly by restraining the construction to a maximal depth h11h-1italic_h - 1 instead of hhitalic_h. In principle, we could then apply the analytic treatment of Section 4 to the system of the bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. However we feel that the existence of a recursive structure of the system of the bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT with a well-defined K(l0,..,lL)K(l_{0},..,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , . . , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) for each term in the recursive description is slightly less clear. Therefore we preferred to work with the aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT which have a well-defined K(ai)𝐾subscript𝑎𝑖K(a_{i})italic_K ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ). In Appendix A we will discuss another algorithm that yields in general even more compact systems of equations.

Refer to caption
Figure 5. The classes corresponding to the bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of equations (3)

3.2. From Planted Rooted Trees to Rooted and Unrooted Trees

The next step is to find equations for the exponential generating function of rooted trees (where occurrences of the pattern are marked with u𝑢uitalic_u). As above we set

r(x,u)=n,m=0rn,mxnumn!,𝑟𝑥𝑢superscriptsubscript𝑛𝑚0subscript𝑟𝑛𝑚superscript𝑥𝑛superscript𝑢𝑚𝑛r(x,u)=\sum_{n,m=0}^{\infty}r_{n,m}\frac{x^{n}u^{m}}{n!},italic_r ( italic_x , italic_u ) = ∑ start_POSTSUBSCRIPT italic_n , italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG ,

where rn,msubscript𝑟𝑛𝑚r_{n,m}italic_r start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT denotes the number of rooted trees of size n𝑛nitalic_n with exactly m𝑚mitalic_m occurrences of the pattern {\mathcal{M}}caligraphic_M. (That is, occurrences of the rooted patterns rsubscript𝑟{\mathcal{M}}_{r}caligraphic_M start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT deducible from {\mathcal{M}}caligraphic_M. Here, a rooted pattern is defined in a very similar way as a planted pattern.)

Proposition 2.

(Rooted Trees) Let {\mathcal{M}}caligraphic_M be a pattern and let

a0(x,u),,aL(x,u)subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢a_{0}(x,u),\ldots,a_{L}(x,u)italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u )

denote the auxiliary functions introduced in Proposition 1. Then there exists a polynomial Q(y0,,yL,u)𝑄subscript𝑦0normal-…subscript𝑦𝐿𝑢Q(y_{0},\ldots,y_{L},u)italic_Q ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) with non-negative coefficients satisfying Q(y0,,yL,1)cey0++yLsubscript𝑐𝑄subscript𝑦0normal-…subscript𝑦𝐿1superscript𝑒subscript𝑦0normal-⋯subscript𝑦𝐿Q(y_{0},\ldots,y_{L},1)\leq_{c}e^{y_{0}+\cdots+y_{L}}italic_Q ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , 1 ) ≤ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, and such that

(4) r(x,u)=G(x,u,a0(x,u),,aL(x,u))𝑟𝑥𝑢𝐺𝑥𝑢subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢r(x,u)=G(x,u,a_{0}(x,u),\ldots,a_{L}(x,u))italic_r ( italic_x , italic_u ) = italic_G ( italic_x , italic_u , italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) )

for

(5) G(x,u,y0,,yL)=x(ey0++yLQ(y0,,yL,1)+Q(y0,,yL,u)).𝐺𝑥𝑢subscript𝑦0subscript𝑦𝐿𝑥superscript𝑒subscript𝑦0subscript𝑦𝐿𝑄subscript𝑦0subscript𝑦𝐿1𝑄subscript𝑦0subscript𝑦𝐿𝑢G(x,u,y_{0},\ldots,y_{L})=x\left(e^{y_{0}+\dots+y_{L}}-Q(y_{0},\ldots,y_{L},1)+Q(y_{0},\ldots,y_{L},u)\right).italic_G ( italic_x , italic_u , italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) = italic_x ( italic_e start_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - italic_Q ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , 1 ) + italic_Q ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) ) .
Proof.

The proof is in principle a direct continuation of the proof of Proposition 1. We recall that a rooted tree is just a root with zero, one, two, \dots planted subtrees, i.e., the class of rooted trees can be described as a disjoint union of classes c𝑐citalic_c of rooted trees of the form xaj1ajd𝑥subscript𝑎subscript𝑗1subscript𝑎subscript𝑗𝑑xa_{j_{1}}\dotsm a_{j_{d}}italic_x italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Furthermore, let lisubscript𝑙𝑖l_{i}italic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT denote the number of classes aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in this term such that c=xa0l0aLlL𝑐𝑥superscriptsubscript𝑎0subscript𝑙0superscriptsubscript𝑎𝐿subscript𝑙𝐿c=xa_{0}^{l_{0}}\dotsm a_{L}^{l_{L}}italic_c = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, and set K¯(l0,,lL)¯𝐾subscript𝑙0subscript𝑙𝐿\bar{K}(l_{0},\ldots,l_{L})over¯ start_ARG italic_K end_ARG ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) to be the number of additional occurrences of the pattern {\mathcal{M}}{}caligraphic_M. This number again corresponds to the number of occurrences of {\mathcal{M}}caligraphic_M in a (rooted) tree Tc𝑇𝑐T\in citalic_T ∈ italic_c in which the root of T𝑇Titalic_T occurs as internal node of the pattern. Set

Qd(y0,,yL,u)=l0++lL=d1l0!lL!y0l0yLlLuK¯(l0,,lL).subscript𝑄𝑑subscript𝑦0subscript𝑦𝐿𝑢subscriptsubscript𝑙0subscript𝑙𝐿𝑑1subscript𝑙0subscript𝑙𝐿superscriptsubscript𝑦0subscript𝑙0superscriptsubscript𝑦𝐿subscript𝑙𝐿superscript𝑢¯𝐾subscript𝑙0subscript𝑙𝐿Q_{d}(y_{0},\ldots,y_{L},u)=\sum_{l_{0}+\dots+l_{L}=d}\frac{1}{l_{0}!\cdots l_{L}!}y_{0}^{l_{0}}\cdots y_{L}^{l_{L}}u^{\bar{K}(l_{0},\ldots,l_{L})}.italic_Q start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) = ∑ start_POSTSUBSCRIPT italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = italic_d end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ! ⋯ italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ! end_ARG italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT over¯ start_ARG italic_K end_ARG ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT .

Then by construction

r(x,u)=xd0Qd(a0(x,u),,aL(x,u),u).𝑟𝑥𝑢𝑥subscript𝑑0subscript𝑄𝑑subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢𝑢r(x,u)=x\sum_{d\geq 0}Q_{d}(a_{0}(x,u),\ldots,a_{L}(x,u),u).italic_r ( italic_x , italic_u ) = italic_x ∑ start_POSTSUBSCRIPT italic_d ≥ 0 end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) , italic_u ) .

Note that d0Qd(y0,,yL,1)=ey0++yLsubscript𝑑0subscript𝑄𝑑subscript𝑦0subscript𝑦𝐿1superscript𝑒subscript𝑦0subscript𝑦𝐿\sum_{d\geq 0}Q_{d}(y_{0},\ldots,y_{L},1)=e^{y_{0}+\cdots+y_{L}}∑ start_POSTSUBSCRIPT italic_d ≥ 0 end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , 1 ) = italic_e start_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. Let D¯¯𝐷\bar{D}over¯ start_ARG italic_D end_ARG denote the set of degrees of the internal (filled) nodes of the pattern, that is, D¯={d+1:dD}¯𝐷conditional-set𝑑1𝑑𝐷\bar{D}=\{\,d+1:d\in D\,\}over¯ start_ARG italic_D end_ARG = { italic_d + 1 : italic_d ∈ italic_D }; then Qd(y0,,yL,u)subscript𝑄𝑑subscript𝑦0subscript𝑦𝐿𝑢Q_{d}(y_{0},\ldots,y_{L},u)italic_Q start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) does not depend on u𝑢uitalic_u if dD¯𝑑¯𝐷d\not\in\bar{D}italic_d ∉ over¯ start_ARG italic_D end_ARG. With

Q(y0,,yL,u):=dD¯Qd(y0,,yL,u),assign𝑄subscript𝑦0subscript𝑦𝐿𝑢subscript𝑑¯𝐷subscript𝑄𝑑subscript𝑦0subscript𝑦𝐿𝑢Q(y_{0},\ldots,y_{L},u):=\sum_{d\in\bar{D}}Q_{d}(y_{0},\ldots,y_{L},u),italic_Q ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) := ∑ start_POSTSUBSCRIPT italic_d ∈ over¯ start_ARG italic_D end_ARG end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) ,

we obtain (4) and (5). The number K¯(l0,,lL)¯𝐾subscript𝑙0subscript𝑙𝐿\bar{K}(l_{0},\dots,l_{L})over¯ start_ARG italic_K end_ARG ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) is well-defined for a similar reason as was K(l0,,lL)𝐾subscript𝑙0subscript𝑙𝐿K(l_{0},\dots,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ), and can be calculated similarly. ∎

Refer to caption
Figure 6. Rooted patterns for the pattern in Figure 1

We again illustrate the proof with our example. In Figure 6 the corresponding rooted patterns are shown. For convenience let r0=r0(x,u)subscript𝑟0subscript𝑟0𝑥𝑢r_{0}=r_{0}(x,u)italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) denote the function

r0=xepxp33!,subscript𝑟0𝑥superscript𝑒𝑝𝑥superscript𝑝33r_{0}=xe^{p}-\frac{xp^{3}}{3!},italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x italic_e start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - divide start_ARG italic_x italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG start_ARG 3 ! end_ARG ,

where p=a0++a10𝑝subscript𝑎0subscript𝑎10p=a_{0}+\cdots+a_{10}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT. The function r0subscript𝑟0r_{0}italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT might also be interpreted as a catch-all function for the “uninteresting” subtrees—just a root x𝑥xitalic_x with an unspecified number of planted trees attached, except the ones we handle differently, namely the cases dD¯={3}𝑑¯𝐷3d\in\bar{D}=\{3\}italic_d ∈ over¯ start_ARG italic_D end_ARG = { 3 }. The generating function r=r(x,u)𝑟𝑟𝑥𝑢r=r(x,u)italic_r = italic_r ( italic_x , italic_u ) for rooted trees is then given by

r=r0+16xb03+12x1i3b02biui1+12x1i,j3b0bibjui+j1+16x1i,j,k3bibjbkui+j+k𝑟subscript𝑟016𝑥superscriptsubscript𝑏0312𝑥subscript1𝑖3superscriptsubscript𝑏02subscript𝑏𝑖superscript𝑢𝑖112𝑥subscriptformulae-sequence1𝑖𝑗3subscript𝑏0subscript𝑏𝑖subscript𝑏𝑗superscript𝑢𝑖𝑗116𝑥subscriptformulae-sequence1𝑖𝑗𝑘3subscript𝑏𝑖subscript𝑏𝑗subscript𝑏𝑘superscript𝑢𝑖𝑗𝑘r=r_{0}+\frac{1}{6}xb_{0}^{3}+\frac{1}{2}x\sum_{1\leq i\leq 3}b_{0}^{2}b_{i}u^{i-1}+\frac{1}{2}x\sum_{1\leq i,j\leq 3}b_{0}b_{i}b_{j}u^{i+j-1}+\frac{1}{6}x\sum_{1\leq i,j,k\leq 3}b_{i}b_{j}b_{k}u^{i+j+k}italic_r = italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 6 end_ARG italic_x italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x ∑ start_POSTSUBSCRIPT 1 ≤ italic_i ≤ 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT italic_i - 1 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x ∑ start_POSTSUBSCRIPT 1 ≤ italic_i , italic_j ≤ 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT italic_i + italic_j - 1 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 6 end_ARG italic_x ∑ start_POSTSUBSCRIPT 1 ≤ italic_i , italic_j , italic_k ≤ 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT italic_i + italic_j + italic_k end_POSTSUPERSCRIPT

where the bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are defined in (3).

As above we have tn,m=rn,m/nsubscript𝑡𝑛𝑚subscript𝑟𝑛𝑚𝑛t_{n,m}=r_{n,m}/nitalic_t start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT / italic_n, where tn,msubscript𝑡𝑛𝑚t_{n,m}italic_t start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT denotes the number of unrooted trees with n𝑛nitalic_n nodes and exactly m𝑚mitalic_m occurrences of the pattern {\mathcal{M}}caligraphic_M. This relation is sufficient for our purposes. It is also possible to express the corresponding generating function t(x,u)𝑡𝑥𝑢t(x,u)italic_t ( italic_x , italic_u ). In a way similar as before, we can define the number of additional occurrences K^(i,j)^𝐾𝑖𝑗\hat{K}(i,j)over^ start_ARG italic_K end_ARG ( italic_i , italic_j ) of the pattern {\mathcal{M}}caligraphic_M that appear by constructing an unrooted tree from two planted trees of the class aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT by identifying the additional edges at their planted roots and discarding the phantom nodes. For our example we get

t(x,u)=r(x,u)12p(x,u)2121i,j3bi(x,u)bj(x,u)(ui+j21).𝑡𝑥𝑢𝑟𝑥𝑢12𝑝superscript𝑥𝑢212subscriptformulae-sequence1𝑖𝑗3subscript𝑏𝑖𝑥𝑢subscript𝑏𝑗𝑥𝑢superscript𝑢𝑖𝑗21t(x,u)=r(x,u)-\frac{1}{2}p(x,u)^{2}-\frac{1}{2}\sum_{1\leq i,j\leq 3}b_{i}(x,u)b_{j}(x,u)(u^{i+j-2}-1).italic_t ( italic_x , italic_u ) = italic_r ( italic_x , italic_u ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_p ( italic_x , italic_u ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT 1 ≤ italic_i , italic_j ≤ 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_u ) italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , italic_u ) ( italic_u start_POSTSUPERSCRIPT italic_i + italic_j - 2 end_POSTSUPERSCRIPT - 1 ) .

4. Asymptotic Behavior

Since we are not interested in the actual number of occurrences of the pattern, but only in its asymptotic behavior, we do not have to compute explicit formulae from the system of equations. Instead, we apply a result slightly adapted from [Drm97] which we state and discuss in Appendix B. In fact, it is immediately clear that Theorem 2 in this appendix, whose object is the proof of Gaussian limiting distributions, applies to the kind of problem we are interested in: the assertions of Propositions 1 and 2 exactly fit the assumptions of Theorem 2.

The only missing point is the existence of a non-negative solution (x0,𝐚0)subscript𝑥0subscript𝐚0(x_{0},{\bf a}_{0})( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) of the system

(6) 𝐚𝐚\displaystyle{\bf a}bold_a =𝐅(x,𝐚,1),absent𝐅𝑥𝐚1\displaystyle={\bf F}(x,{\bf a},{1}),= bold_F ( italic_x , bold_a , 1 ) ,
(7) 00\displaystyle 0 =det(𝐈𝐅𝐚(x,𝐚,1)),absent𝐈subscript𝐅𝐚𝑥𝐚1\displaystyle=\det({\bf I}-{\bf F}_{\bf a}(x,{\bf a},{1})),= roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x , bold_a , 1 ) ) ,

where (6) is the system of functional equations of Proposition 1 and 𝐅𝐚subscript𝐅𝐚{\bf F}_{\bf a}bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT is the Jacobian matrix of 𝐅𝐅{\bf F}bold_F. Since the sum of all unknown functions p(x,u)𝑝𝑥𝑢p(x,u)italic_p ( italic_x , italic_u ) is known for u=1𝑢1u=1italic_u = 1:

p(x,1)=p(x)=n1nn1xnn!=121ex+,𝑝𝑥1𝑝𝑥subscript𝑛1superscript𝑛𝑛1superscript𝑥𝑛𝑛121𝑒𝑥p(x,1)=p(x)=\sum_{n\geq 1}n^{n-1}\frac{x^{n}}{n!}=1-\sqrt{2}\sqrt{1-ex}+\cdots,italic_p ( italic_x , 1 ) = italic_p ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n ≥ 1 end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT divide start_ARG italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG italic_n ! end_ARG = 1 - square-root start_ARG 2 end_ARG square-root start_ARG 1 - italic_e italic_x end_ARG + ⋯ ,

it is not unexpected that x0=1/esubscript𝑥01𝑒x_{0}=1/eitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e.

Proposition 3.

There exists a unique non-negative solution (x0,𝐚0)subscript𝑥0subscript𝐚0(x_{0},{\bf a}_{0})( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) of System (67), for which x0=1/esubscript𝑥01𝑒x_{0}=1/eitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e and the components of 𝐚0subscript𝐚0{\bf a}_{0}bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT are polynomials (with rational coefficients) in 1/e1𝑒1/e1 / italic_e.

Proof.

For a proof, set u=1𝑢1u=1italic_u = 1 and consider the solution 𝐚(x,1)=(a0(x,1),,aL1(x,1))𝐚𝑥1subscript𝑎0𝑥1subscript𝑎𝐿1𝑥1{\bf a}(x,1)=(a_{0}(x,1),\ldots,a_{L-1}(x,1))bold_a ( italic_x , 1 ) = ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) , … , italic_a start_POSTSUBSCRIPT italic_L - 1 end_POSTSUBSCRIPT ( italic_x , 1 ) ). Since the dependency graph is strongly connected it follows that all functions aj(x,1)subscript𝑎𝑗𝑥1a_{j}(x,1)italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , 1 ) have the same radius of convergence which has to be x0=1/esubscript𝑥01𝑒x_{0}=1/eitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e, and all functions are singular at x=x0𝑥subscript𝑥0x=x_{0}italic_x = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Since 0aj(x,1)p(x,1)<0subscript𝑎𝑗𝑥1𝑝𝑥10\leq a_{j}(x,1)\leq p(x,1)<\infty0 ≤ italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , 1 ) ≤ italic_p ( italic_x , 1 ) < ∞ for 0xx00𝑥subscript𝑥00\leq x\leq x_{0}0 ≤ italic_x ≤ italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT it also follows that aj(x0,1)subscript𝑎𝑗subscript𝑥01a_{j}(x_{0},1)italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) is finite, and we have 𝐚(x0,1)=𝐅(x0,𝐚(x0,1),1)𝐚subscript𝑥01𝐅subscript𝑥0𝐚subscript𝑥011{\bf a}(x_{0},1)={\bf F}(x_{0},{\bf a}(x_{0},1),1)bold_a ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) = bold_F ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) , 1 ). If we had the inequality det(𝐈𝐅𝐚(x0,𝐚(x0,1),1))0𝐈subscript𝐅𝐚subscript𝑥0𝐚subscript𝑥0110\det({\bf I}-{\bf F}_{\bf a}(x_{0},{\bf a}(x_{0},1),{1}))\neq 0roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) , 1 ) ) ≠ 0 then the implicit function theorem would imply the existence of an analytic continuation for aj(x,1)subscript𝑎𝑗𝑥1a_{j}(x,1)italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , 1 ) around x=x0𝑥subscript𝑥0x=x_{0}italic_x = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, which is, of course, a contradiction. Thus, the determinant is zero and system (67) has a unique solution.

To see that the components a¯0,,a¯Lsubscript¯𝑎0subscript¯𝑎𝐿\bar{a}_{0},\dots,\bar{a}_{L}over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT (with a¯i=ai(1/e,1)subscript¯𝑎𝑖subscript𝑎𝑖1𝑒1\bar{a}_{i}=a_{i}(1/e,1)over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 / italic_e , 1 )) of 𝐚0subscript𝐚0{\bf a}_{0}bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT are polynomials in 1/e1𝑒1/e1 / italic_e we will construct the partition 𝒜={a0,a1,,aL}𝒜subscript𝑎0subscript𝑎1subscript𝑎𝐿\mathcal{A}=\{a_{0},a_{1},\dots,a_{L}\}caligraphic_A = { italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT } on which the system of equations (67) is based by refining step by step the trivial partition consisting of only one class p𝑝pitalic_p. The recursive description of this trivial partition is given by the formal equation p=xi0pi𝑝𝑥subscript𝑖0superscript𝑝𝑖p=x\sum_{i\geq 0}p^{i}italic_p = italic_x ∑ start_POSTSUBSCRIPT italic_i ≥ 0 end_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT. Additionally, the solution of the corresponding equation p=xexp(p)𝑝𝑥𝑝p=x\exp(p)italic_p = italic_x roman_exp ( italic_p ) for the generating function p𝑝pitalic_p (denoted by the same symbol p𝑝pitalic_p) is given by (x0,p¯)=(1/e,1)subscript𝑥0¯𝑝1𝑒1(x_{0},\bar{p})=(1/e,1)( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over¯ start_ARG italic_p end_ARG ) = ( 1 / italic_e , 1 ), with p¯¯𝑝\bar{p}over¯ start_ARG italic_p end_ARG clearly a (constant) polynomial in 1/e1𝑒1/e1 / italic_e. Now let D={d1,,ds}(s)𝐷subscript𝑑1subscript𝑑𝑠𝑠D=\{d_{1},\dots,d_{s}\}\ (s\in\mathbb{N})italic_D = { italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_d start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT } ( italic_s ∈ blackboard_N ) again denote the set of out-degrees that occur in the planted patterns. We will refine p𝑝pitalic_p by introducing for each diDsubscript𝑑𝑖𝐷d_{i}\in Ditalic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_D a class aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT consisting of all trees of root out-degree disubscript𝑑𝑖d_{i}italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, as well as a class a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT for trees with root out-degree not in D𝐷Ditalic_D. The partition {a0,a1,,as}subscript𝑎0subscript𝑎1subscript𝑎𝑠\{a_{0},a_{1},\dots,a_{s}\}{ italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT } has the recursive description

a0subscript𝑎0\displaystyle a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =xjD(a0a1as)j,absent𝑥subscript𝑗𝐷superscriptdirect-sumsubscript𝑎0subscript𝑎1subscript𝑎𝑠𝑗\displaystyle=x\sum_{j\in\mathbb{N}\setminus D}(a_{0}\oplus a_{1}\oplus\cdots\oplus a_{s})^{j},= italic_x ∑ start_POSTSUBSCRIPT italic_j ∈ blackboard_N ∖ italic_D end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ ⋯ ⊕ italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ,
(8) aisubscript𝑎𝑖\displaystyle a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT =x(a0a1as)di(i=1,,s),absent𝑥superscriptdirect-sumsubscript𝑎0subscript𝑎1subscript𝑎𝑠subscript𝑑𝑖𝑖1𝑠\displaystyle=x(a_{0}\oplus a_{1}\oplus\cdots\oplus a_{s})^{d_{i}}\qquad(i=1,\dots,s),= italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ ⋯ ⊕ italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_i = 1 , … , italic_s ) ,

and the solution of the corresponding system of equations

a0(x,1)subscript𝑎0𝑥1\displaystyle a_{0}(x,1)italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) =xjD1j!(a0(x,1)+a1(x,1)++as(x,1))jabsent𝑥subscript𝑗𝐷1𝑗superscriptsubscript𝑎0𝑥1subscript𝑎1𝑥1subscript𝑎𝑠𝑥1𝑗\displaystyle=x\sum_{j\in\mathbb{N}\setminus D}\frac{1}{j!}(a_{0}(x,1)+a_{1}(x,1)+\cdots+a_{s}(x,1))^{j}= italic_x ∑ start_POSTSUBSCRIPT italic_j ∈ blackboard_N ∖ italic_D end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_j ! end_ARG ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) + ⋯ + italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_x , 1 ) ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT
=xea0(x,1)++as(x,1)xi=1s1di!(a0(x,1)+a1(x,1)++as(x,1))diabsent𝑥superscript𝑒subscript𝑎0𝑥1subscript𝑎𝑠𝑥1𝑥superscriptsubscript𝑖1𝑠1subscript𝑑𝑖superscriptsubscript𝑎0𝑥1subscript𝑎1𝑥1subscript𝑎𝑠𝑥1subscript𝑑𝑖\displaystyle=xe^{a_{0}(x,1)+\cdots+a_{s}(x,1)}-x\sum_{i=1}^{s}\frac{1}{d_{i}!}(a_{0}(x,1)+a_{1}(x,1)+\cdots+a_{s}(x,1))^{d_{i}}= italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) + ⋯ + italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_x , 1 ) end_POSTSUPERSCRIPT - italic_x ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ! end_ARG ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) + ⋯ + italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_x , 1 ) ) start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT
(9) =xep(x)xi=1s1di!p(x)di,absent𝑥superscript𝑒𝑝𝑥𝑥superscriptsubscript𝑖1𝑠1subscript𝑑𝑖𝑝superscript𝑥subscript𝑑𝑖\displaystyle=xe^{p(x)}-x\sum_{i=1}^{s}\frac{1}{d_{i}!}p(x)^{d_{i}},= italic_x italic_e start_POSTSUPERSCRIPT italic_p ( italic_x ) end_POSTSUPERSCRIPT - italic_x ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ! end_ARG italic_p ( italic_x ) start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ,
ai(x,1)subscript𝑎𝑖𝑥1\displaystyle a_{i}(x,1)italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , 1 ) =xdi!(a0(x,1)+a1(x,1)++as(x,1))di=xdi!p(x)di(i=1,,s),formulae-sequenceabsent𝑥subscript𝑑𝑖superscriptsubscript𝑎0𝑥1subscript𝑎1𝑥1subscript𝑎𝑠𝑥1subscript𝑑𝑖𝑥subscript𝑑𝑖𝑝superscript𝑥subscript𝑑𝑖𝑖1𝑠\displaystyle=\frac{x}{d_{i}!}(a_{0}(x,1)+a_{1}(x,1)+\cdots+a_{s}(x,1))^{d_{i}}=\frac{x}{d_{i}!}p(x)^{d_{i}}\qquad(i=1,\dots,s),= divide start_ARG italic_x end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ! end_ARG ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) + ⋯ + italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_x , 1 ) ) start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = divide start_ARG italic_x end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ! end_ARG italic_p ( italic_x ) start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_i = 1 , … , italic_s ) ,

is given by

(10) x0=1/e,a¯i=1di!e(i=1,,s),a¯0=1(a¯1++a¯s),formulae-sequencesubscript𝑥01𝑒formulae-sequencesubscript¯𝑎𝑖1subscript𝑑𝑖𝑒𝑖1𝑠subscript¯𝑎01subscript¯𝑎1subscript¯𝑎𝑠x_{0}=1/e,\qquad\bar{a}_{i}=\frac{1}{d_{i}!\,e}\quad(i=1,\dots,s),\qquad\bar{a}_{0}=1-(\bar{a}_{1}+\cdots+\bar{a}_{s}),italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e , over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ! italic_e end_ARG ( italic_i = 1 , … , italic_s ) , over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 - ( over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ,

thus again polynomials in 1/e1𝑒1/e1 / italic_e. We continue by refining this last partition by introducing classes c1,,cmsubscript𝑐1subscript𝑐𝑚c_{1},\dots,c_{m}italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT (for some m𝑚m\in\mathbb{N}italic_m ∈ blackboard_N) for each term at the right-hand side of (8) after expanding the “multinomial”. Such a class cjsubscript𝑐𝑗c_{j}italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is of the form cj=xa0l0(j)a1l1(j)asls(j)subscript𝑐𝑗𝑥superscriptsubscript𝑎0superscriptsubscript𝑙0𝑗superscriptsubscript𝑎1superscriptsubscript𝑙1𝑗superscriptsubscript𝑎𝑠superscriptsubscript𝑙𝑠𝑗c_{j}=xa_{0}^{l_{0}^{(j)}}a_{1}^{l_{1}^{(j)}}\cdots a_{s}^{l_{s}^{(j)}}italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT with natural numbers li(j),i=0,,sformulae-sequencesuperscriptsubscript𝑙𝑖𝑗𝑖0𝑠l_{i}^{(j)},\ i=0,\dots,sitalic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT , italic_i = 0 , … , italic_s. We get a new partition {a0,c1,,cm}subscript𝑎0subscript𝑐1subscript𝑐𝑚\{a_{0},c_{1},\dots,c_{m}\}{ italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } which has a recursive description by construction (because we can replace the aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT by disjoint unions of certain cjsubscript𝑐𝑗c_{j}italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT). The corresponding system of equations for the generating functions is given by

cj(x,1)=xl0(j)!l1(j)!ls(j)!a0(x,1)l0(j)a1(x,1)l1(j)as(x,1)ls(j)(j=1,,u)subscript𝑐𝑗𝑥1𝑥superscriptsubscript𝑙0𝑗superscriptsubscript𝑙1𝑗superscriptsubscript𝑙𝑠𝑗subscript𝑎0superscript𝑥1superscriptsubscript𝑙0𝑗subscript𝑎1superscript𝑥1superscriptsubscript𝑙1𝑗subscript𝑎𝑠superscript𝑥1superscriptsubscript𝑙𝑠𝑗𝑗1𝑢c_{j}(x,1)=\frac{x}{l_{0}^{(j)}!\,l_{1}^{(j)}!\cdots l_{s}^{(j)}!}a_{0}(x,1)^{l_{0}^{(j)}}a_{1}(x,1)^{l_{1}^{(j)}}\cdots a_{s}(x,1)^{l_{s}^{(j)}}\quad(j=1,\dots,u)italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , 1 ) = divide start_ARG italic_x end_ARG start_ARG italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ! italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ! ⋯ italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ! end_ARG italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_x , 1 ) start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_j = 1 , … , italic_u )

and consequently we have for x0=1/esubscript𝑥01𝑒x_{0}=1/eitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e the solution

c¯j=1e1l0(j)!l1(j)!ls(j)!a¯0l0(j)a¯1l1(j)a¯sls(j)(j=1,,m)subscript¯𝑐𝑗1𝑒1superscriptsubscript𝑙0𝑗superscriptsubscript𝑙1𝑗superscriptsubscript𝑙𝑠𝑗superscriptsubscript¯𝑎0superscriptsubscript𝑙0𝑗superscriptsubscript¯𝑎1superscriptsubscript𝑙1𝑗superscriptsubscript¯𝑎𝑠superscriptsubscript𝑙𝑠𝑗𝑗1𝑚\bar{c}_{j}=\frac{1}{e}\frac{1}{l_{0}^{(j)}!\,l_{1}^{(j)}!\cdots l_{s}^{(j)}!}\bar{a}_{0}^{l_{0}^{(j)}}\bar{a}_{1}^{l_{1}^{(j)}}\cdots\bar{a}_{s}^{l_{s}^{(j)}}\quad(j=1,\dots,m)over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_e end_ARG divide start_ARG 1 end_ARG start_ARG italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ! italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ! ⋯ italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ! end_ARG over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⋯ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_j = 1 , … , italic_m )

with the a¯isubscript¯𝑎𝑖\bar{a}_{i}over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of (10). Thus the c¯jsubscript¯𝑐𝑗\bar{c}_{j}over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT are again polynomials in 1/e1𝑒1/e1 / italic_e. By continuing this procedure until level hhitalic_h (i.e., performing the refinement step hhitalic_h times) we end up with the partition 𝒜𝒜\mathcal{A}caligraphic_A and we see that the solution for the corresponding system of equations consists of polynomials in 1/e1𝑒1/e1 / italic_e, which completes the proof of Proposition 3. ∎

Note that there is a close link with Galton–Watson branching processes. Let pk=1k!esubscript𝑝𝑘1𝑘𝑒p_{k}=\frac{1}{k!\,e}italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_k ! italic_e end_ARG denote a Poisson offspring distribution. Now we interpret a class aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT as the class of process realizations for which the (non-planar) branching structure at the beginning of the processes corresponds to the root structure of aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Then a¯i=ai(1/e,1)subscript¯𝑎𝑖subscript𝑎𝑖1𝑒1\bar{a}_{i}=a_{i}(1/e,1)over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) is just the probability of this event.

We now solve the system of equations obtained for the example pattern. We have x0=1/esubscript𝑥01𝑒x_{0}=1/eitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e. The components of 𝐚0subscript𝐚0{\bf a}_{0}bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT can easily be obtained by following the construction of the proof of Proposition 4 (or we use the branching process interpretation). For example, if we set p=1/(2e)𝑝12𝑒p=1/(2e)italic_p = 1 / ( 2 italic_e ) for the probability of an out-degree 2222 and q=1p𝑞1𝑝q=1-pitalic_q = 1 - italic_p then we get a¯4=a4(1/e,1)=2qp3=2e116e5subscript¯𝑎4subscript𝑎41𝑒12𝑞superscript𝑝32𝑒116superscript𝑒5\bar{a}_{4}=a_{4}(1/e,1)=2qp^{3}=\frac{2e-1}{16e^{5}}over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) = 2 italic_q italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT = divide start_ARG 2 italic_e - 1 end_ARG start_ARG 16 italic_e start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG. The factor 2222 comes from the fact that the two subtrees of the root may be interchanged, see Figure 4. The other classes can be treated similarly and we find:

(11) p(1/e,1)𝑝1𝑒1\displaystyle p(1/e,1)italic_p ( 1 / italic_e , 1 ) =1,absent1\displaystyle=1,= 1 , a5(1/e,1)subscript𝑎51𝑒1\displaystyle a_{5}(1/e,1)italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)4/(128e7),absentsuperscript2𝑒14128superscript𝑒7\displaystyle={(2e-1)^{4}}/{(128e^{7})},= ( 2 italic_e - 1 ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT / ( 128 italic_e start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ) ,
a0(1/e,1)subscript𝑎01𝑒1\displaystyle a_{0}(1/e,1)italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)/(2e),absent2𝑒12𝑒\displaystyle={(2e-1)}/{(2e)},= ( 2 italic_e - 1 ) / ( 2 italic_e ) , a6(1/e,1)subscript𝑎61𝑒1\displaystyle a_{6}(1/e,1)italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)3/(32e7),absentsuperscript2𝑒1332superscript𝑒7\displaystyle={(2e-1)^{3}}/{(32e^{7})},= ( 2 italic_e - 1 ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / ( 32 italic_e start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ) ,
a1(1/e,1)subscript𝑎11𝑒1\displaystyle a_{1}(1/e,1)italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)2/(8e3),absentsuperscript2𝑒128superscript𝑒3\displaystyle={(2e-1)^{2}}/{(8e^{3})},= ( 2 italic_e - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 8 italic_e start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) , a7(1/e,1)subscript𝑎71𝑒1\displaystyle a_{7}(1/e,1)italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)2/(64e7),absentsuperscript2𝑒1264superscript𝑒7\displaystyle={(2e-1)^{2}}/{(64e^{7})},= ( 2 italic_e - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 64 italic_e start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ) ,
a2(1/e,1)subscript𝑎21𝑒1\displaystyle a_{2}(1/e,1)italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)3/(16e5),absentsuperscript2𝑒1316superscript𝑒5\displaystyle={(2e-1)^{3}}/{(16e^{5})},= ( 2 italic_e - 1 ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / ( 16 italic_e start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT ) , a8(1/e,1)subscript𝑎81𝑒1\displaystyle a_{8}(1/e,1)italic_a start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)2/(32e7),absentsuperscript2𝑒1232superscript𝑒7\displaystyle={(2e-1)^{2}}/{(32e^{7})},= ( 2 italic_e - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 32 italic_e start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ) ,
a3(1/e,1)subscript𝑎31𝑒1\displaystyle a_{3}(1/e,1)italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)2/(8e5),absentsuperscript2𝑒128superscript𝑒5\displaystyle={(2e-1)^{2}}/{(8e^{5})},= ( 2 italic_e - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 8 italic_e start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT ) , a9(1/e,1)subscript𝑎91𝑒1\displaystyle a_{9}(1/e,1)italic_a start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)/(32e7),absent2𝑒132superscript𝑒7\displaystyle={(2e-1)}/{(32e^{7})},= ( 2 italic_e - 1 ) / ( 32 italic_e start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ) ,
a4(1/e,1)subscript𝑎41𝑒1\displaystyle a_{4}(1/e,1)italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =(2e1)/(16e5),absent2𝑒116superscript𝑒5\displaystyle={(2e-1)}/{(16e^{5})},= ( 2 italic_e - 1 ) / ( 16 italic_e start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT ) , a10(1/e,1)subscript𝑎101𝑒1\displaystyle a_{10}(1/e,1)italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( 1 / italic_e , 1 ) =1/(128e7).absent1128superscript𝑒7\displaystyle={1}/{(128e^{7})}.= 1 / ( 128 italic_e start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ) .

We are now ready to complete the proof of the main part of Theorem 1. By Propositions 13 we can apply Theorem 2 and it follows that the numbers rn,msubscript𝑟𝑛𝑚r_{n,m}italic_r start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT have a Gaussian limiting distribution with mean and variance which are proportional to n𝑛nitalic_n. Since tn,m=rn,m/nsubscript𝑡𝑛𝑚subscript𝑟𝑛𝑚𝑛t_{n,m}=r_{n,m}/nitalic_t start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT / italic_n we get exactly the same law for unrooted trees. It remains to compute μ𝜇\muitalic_μ and σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

By using the procedure described in Appendix B we get for our expample pattern

μ=58e3=0.0311169177𝜇58superscript𝑒30.0311169177\mu=\frac{5}{8e^{3}}=0.0311169177\dotsitalic_μ = divide start_ARG 5 end_ARG start_ARG 8 italic_e start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG = 0.0311169177 …

and

σ2=20e3+72e2+84e17532e6=0.0764585401.superscript𝜎220superscript𝑒372superscript𝑒284𝑒17532superscript𝑒60.0764585401\sigma^{2}=\frac{20e^{3}+72e^{2}+84e-175}{32e^{6}}=0.0764585401\dots.italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 20 italic_e start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 72 italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 84 italic_e - 175 end_ARG start_ARG 32 italic_e start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT end_ARG = 0.0764585401 … .

We observe—as predicted by Theorem 1—that both μ𝜇\muitalic_μ and σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT can be written as rational polynomials in 1/e1𝑒1/e1 / italic_e.

In what follows we will prove this fact (which completes the proof of Theorem 1) and also present an easy formula for μ𝜇\muitalic_μ. Unfortunately the procedure for calculating σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is much more complicated so that it seems that there is no simple formula.

Proposition 4.

Let x0=1/esubscript𝑥01𝑒x_{0}=1/eitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e and 𝐚0subscript𝐚0{\bf a}_{0}bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT be given by Proposition 3 and let Pj(𝐲,u)subscript𝑃𝑗𝐲𝑢P_{j}({\bf y},u)italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( bold_y , italic_u ) (1jL)1𝑗𝐿(1\leq j\leq L)( 1 ≤ italic_j ≤ italic_L ) be the polynomials of Proposition 1, with 𝐲=(y0,,yL)𝐲subscript𝑦0normal-…subscript𝑦𝐿{\bf y}=(y_{0},\dots,y_{L})bold_y = ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ). Then μ𝜇\muitalic_μ (of Theorem 1) is a polynomial in 1/e1𝑒1/e1 / italic_e with rational coefficients and is given by

(12) μ=1ej=1LPju(𝐚0,1).𝜇1𝑒superscriptsubscript𝑗1𝐿subscript𝑃𝑗𝑢subscript𝐚01\mu=\frac{1}{e}\sum_{j=1}^{L}\frac{\partial P_{j}}{\partial u}({\bf a}_{0},1).italic_μ = divide start_ARG 1 end_ARG start_ARG italic_e end_ARG ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT divide start_ARG ∂ italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_u end_ARG ( bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) .
Proof.

Let 𝐚=𝐅(x,𝐚,u)𝐚𝐅𝑥𝐚𝑢{\bf a}={\bf F}(x,{\bf a},u)bold_a = bold_F ( italic_x , bold_a , italic_u ) be the system of functional equations of Proposition 1. In Appendix B the following formula for the mean is derived:

(13) μ=1x0𝐛T𝐅u(x0,𝐚0,1)𝐛T𝐅x(x0,𝐚0,1).𝜇1subscript𝑥0superscript𝐛Tsubscript𝐅𝑢subscript𝑥0subscript𝐚01superscript𝐛Tsubscript𝐅𝑥subscript𝑥0subscript𝐚01\mu=\frac{1}{x_{0}}\frac{{\bf b}^{\mathrm{T}}{\bf F}_{u}(x_{0},{\bf a}_{0},1)}{{\bf b}^{\mathrm{T}}{\bf F}_{x}(x_{0},{\bf a}_{0},1)}.italic_μ = divide start_ARG 1 end_ARG start_ARG italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG divide start_ARG bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) end_ARG start_ARG bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) end_ARG .

Here 𝐛Tsuperscript𝐛𝑇{\bf b}^{T}bold_b start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT  denotes a positive left eigenvector of 𝐈𝐅𝐚𝐈subscript𝐅𝐚{\bf I}-{\bf F}_{\bf a}bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT, which is unique up to scaling.

From the equality

𝐅(x,𝐚,u)=(x(ea0++aLj=1LPj(𝐚,1))xP1(𝐚,u)xP2(𝐚,u)xPL(𝐚,u)),𝐅𝑥𝐚𝑢𝑥superscript𝑒subscript𝑎0subscript𝑎𝐿superscriptsubscript𝑗1𝐿subscript𝑃𝑗𝐚1𝑥subscript𝑃1𝐚𝑢𝑥subscript𝑃2𝐚𝑢𝑥subscript𝑃𝐿𝐚𝑢{\bf F}(x,{\bf a},u)=\left(\begin{array}[]{c}x\left(e^{a_{0}+\cdots+a_{L}}-\sum_{j=1}^{L}P_{j}({\bf a},1)\right)\\ xP_{1}({\bf a},u)\\ xP_{2}({\bf a},u)\\ \vdots\\ xP_{L}({\bf a},u)\end{array}\right),bold_F ( italic_x , bold_a , italic_u ) = ( start_ARRAY start_ROW start_CELL italic_x ( italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( bold_a , 1 ) ) end_CELL end_ROW start_ROW start_CELL italic_x italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( bold_a , italic_u ) end_CELL end_ROW start_ROW start_CELL italic_x italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( bold_a , italic_u ) end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_x italic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( bold_a , italic_u ) end_CELL end_ROW end_ARRAY ) ,

we get, after denoting Piajsubscript𝑃𝑖subscript𝑎𝑗\frac{\partial P_{i}}{\partial a_{j}}divide start_ARG ∂ italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG with Pi,ajsubscript𝑃𝑖subscript𝑎𝑗P_{i,a_{j}}italic_P start_POSTSUBSCRIPT italic_i , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT,

(14) 𝐅𝐚=x(ea0++aLj=1LPj,a0ea0++aLj=1LPj,aLP1,a0P1,aLPL,a0PL,aL).subscript𝐅𝐚𝑥superscript𝑒subscript𝑎0subscript𝑎𝐿superscriptsubscript𝑗1𝐿subscript𝑃𝑗subscript𝑎0superscript𝑒subscript𝑎0subscript𝑎𝐿superscriptsubscript𝑗1𝐿subscript𝑃𝑗subscript𝑎𝐿subscript𝑃1subscript𝑎0subscript𝑃1subscript𝑎𝐿missing-subexpressionsubscript𝑃𝐿subscript𝑎0subscript𝑃𝐿subscript𝑎𝐿{\bf F}_{{\bf a}}=x\left(\begin{array}[]{ccc}e^{a_{0}+\cdots+a_{L}}-\sum_{j=1}^{L}P_{j,a_{0}}&\cdots&e^{a_{0}+\cdots+a_{L}}-\sum_{j=1}^{L}P_{j,a_{L}}\\ P_{1,a_{0}}&\cdots&P_{1,a_{L}}\\ \vdots&&\vdots\\ P_{L,a_{0}}&\cdots&P_{L,a_{L}}\\ \end{array}\right).bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT = italic_x ( start_ARRAY start_ROW start_CELL italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_j , italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_j , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_P start_POSTSUBSCRIPT 1 , italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_P start_POSTSUBSCRIPT 1 , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_P start_POSTSUBSCRIPT italic_L , italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_P start_POSTSUBSCRIPT italic_L , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) .

Since a0(x0,1)++aL(x0,1)=p(x0,1)=1subscript𝑎0subscript𝑥01subscript𝑎𝐿subscript𝑥01𝑝subscript𝑥011a_{0}(x_{0},1)+\cdots+a_{L}(x_{0},1)=p(x_{0},1)=1italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) = italic_p ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) = 1 we have x0ea0(x0,1)+aL(x0,1)=1subscript𝑥0superscript𝑒subscript𝑎0subscript𝑥01subscript𝑎𝐿subscript𝑥011x_{0}e^{a_{0}(x_{0},1)+\cdots a_{L}(x_{0},1)}=1italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) + ⋯ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) end_POSTSUPERSCRIPT = 1. Consequently the sum of all rows of 𝐅𝐚subscript𝐅𝐚{\bf F}_{\bf a}bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT equals (1,1,,1)111(1,1,\ldots,1)( 1 , 1 , … , 1 ) for x=x0=1/e𝑥subscript𝑥01𝑒x=x_{0}=1/eitalic_x = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e. Thus, denoting the transpose of a vector v𝑣vitalic_v by vTsuperscript𝑣Tv^{\mathrm{T}}italic_v start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT, the vector 𝐛T=(1,1,,1)superscript𝐛T111{\bf b}^{\mathrm{T}}=(1,1,\ldots,1)bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT = ( 1 , 1 , … , 1 ) is the unique positive left eigenvector of 𝐈𝐅𝐚𝐈subscript𝐅𝐚{\bf I}-{\bf F}_{\bf a}bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT, up to scaling.

It is now easy to check that

x0𝐛T𝐅x(x0,𝐚0,1)=1eea0(x0,1)+aL(x0,1)=1subscript𝑥0superscript𝐛Tsubscript𝐅𝑥subscript𝑥0subscript𝐚011𝑒superscript𝑒subscript𝑎0subscript𝑥01subscript𝑎𝐿subscript𝑥011x_{0}{\bf b}^{\mathrm{T}}{\bf F}_{x}(x_{0},{\bf a}_{0},1)=\frac{1}{e}e^{a_{0}(x_{0},1)+\cdots a_{L}(x_{0},1)}=1italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) = divide start_ARG 1 end_ARG start_ARG italic_e end_ARG italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) + ⋯ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) end_POSTSUPERSCRIPT = 1

and that

𝐛T𝐅u(x0,𝐚0,1)=1ej=1LPj,u(𝐚0,1).superscript𝐛Tsubscript𝐅𝑢subscript𝑥0subscript𝐚011𝑒superscriptsubscript𝑗1𝐿subscript𝑃𝑗𝑢subscript𝐚01{\bf b}^{\mathrm{T}}{\bf F}_{u}(x_{0},{\bf a}_{0},1)=\frac{1}{e}\sum_{j=1}^{L}P_{j,u}({\bf a}_{0},1).bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) = divide start_ARG 1 end_ARG start_ARG italic_e end_ARG ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_j , italic_u end_POSTSUBSCRIPT ( bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) .

The fact that μ𝜇\muitalic_μ is a polynomial in 1/e1𝑒1/e1 / italic_e is now a direct consequence from the fact that 𝐚0subscript𝐚0{\bf a}_{0}bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT consists of polynomials in 1/e1𝑒1/e1 / italic_e and the fact that the coefficients are rational follows from the fact that 𝐅(x,𝐚,u)𝐅𝑥𝐚𝑢{\bf F}(x,{\bf a},u)bold_F ( italic_x , bold_a , italic_u ) has rational coefficients. ∎

Of course, with help of (12) we can easily evaluate μ𝜇\muitalic_μ directly. As already indicated it seems that there is no simple formula for σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

Before proving Proposition 5 we state in interesting fact that will be used in the sequel.

Lemma 1.

Let a0,a1,,aLsubscript𝑎0subscript𝑎1normal-…subscript𝑎𝐿a_{0},a_{1},\ldots,a_{L}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT the partition of p𝑝pitalic_p that is used in the proof of Theorem 1. Then

det(𝐈𝐅𝐚(x,𝐚,1))=1xea0+a1++aL.𝐈subscript𝐅𝐚𝑥𝐚11𝑥superscript𝑒subscript𝑎0subscript𝑎1subscript𝑎𝐿\det\left({\bf I}-{\bf F}_{\bf a}(x,{\bf a},1)\right)=1-xe^{a_{0}+a_{1}+\cdots+a_{L}}.roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x , bold_a , 1 ) ) = 1 - italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT .

Since the proof is a rather lengthy computation we postpone it to Appendix C.

Proposition 5.

Let x0=1/esubscript𝑥01𝑒x_{0}=1/eitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / italic_e and 𝐚0subscript𝐚0{\bf a}_{0}bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT be given by Proposition 3. Then σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (of Theorem 1) is a polynomial in 1/e1𝑒1/e1 / italic_e (with rational coefficients).

Proof.

From the proof of Proposition 4 we already know that xu(1)subscript𝑥𝑢1x_{u}(1)italic_x start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( 1 ) can be represented as a polynomial in 1/e1𝑒1/e1 / italic_e (with rational coefficients). The next step is to show that 𝐚u(1)subscript𝐚𝑢1{\bf a}_{u}(1)bold_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( 1 ) has the same property. For this purpose we have to look at the system (30)

(𝐈𝐅𝐚)𝐚u𝐈subscript𝐅𝐚subscript𝐚𝑢\displaystyle({\bf I}-{\bf F}_{\bf a}){\bf a}_{u}( bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ) bold_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT =𝐅xxu+𝐅u,absentsubscript𝐅𝑥subscript𝑥𝑢subscript𝐅𝑢\displaystyle={\bf F}_{x}x_{u}+{\bf F}_{u},= bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT + bold_F start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ,
D𝐚𝐚usubscript𝐷𝐚subscript𝐚𝑢\displaystyle-D_{\bf a}{\bf a}_{u}- italic_D start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT bold_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT =Dxxu+Du,absentsubscript𝐷𝑥subscript𝑥𝑢subscript𝐷𝑢\displaystyle=D_{x}x_{u}+D_{u},= italic_D start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ,

where D(x,𝐚,u)=det(𝐈𝐅𝐚(x,𝐚,1))=1xea0+a1++aL𝐷𝑥𝐚𝑢𝐈subscript𝐅𝐚𝑥𝐚11𝑥superscript𝑒subscript𝑎0subscript𝑎1subscript𝑎𝐿D(x,{\bf a},u)=\det\left({\bf I}-{\bf F}_{\bf a}(x,{\bf a},1)\right)=1-xe^{a_{0}+a_{1}+\cdots+a_{L}}italic_D ( italic_x , bold_a , italic_u ) = roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x , bold_a , 1 ) ) = 1 - italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. We first observe that

D𝐚(x0,𝐚0,1)=(1,1,,1).subscript𝐷𝐚subscript𝑥0subscript𝐚01111D_{\bf a}(x_{0},{\bf a}_{0},1)=(-1,-1,\ldots,-1).italic_D start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ) = ( - 1 , - 1 , … , - 1 ) .

Hence, we can replace the first row of the (L+1)×(L+1)𝐿1𝐿1(L+1)\times(L+1)( italic_L + 1 ) × ( italic_L + 1 )-matrix 𝐈𝐅𝐚𝐈subscript𝐅𝐚{\bf I}-{\bf F}_{\bf a}bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT (that is redundant since the matrix has rank L𝐿Litalic_L) by the row (1,1,,1)111(1,1,\ldots,1)( 1 , 1 , … , 1 ) and obtain a regular linear system for 𝐚u(1)subscript𝐚𝑢1{\bf a}_{u}(1)bold_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( 1 ). Note that all entries of the right-hand side of this linear system can be represented as polynomials in 1/e1𝑒1/e1 / italic_e.

Let 𝐌(x,𝐚)𝐌𝑥𝐚{\bf M}(x,{\bf a})bold_M ( italic_x , bold_a ) denote the matrix obtained from 𝐈𝐅𝐚(x,𝐚,1)𝐈subscript𝐅𝐚𝑥𝐚1{\bf I}-{\bf F}_{\bf a}(x,{\bf a},1)bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x , bold_a , 1 ) by replacing the first row by (1,1,,1)111(1,1,\ldots,1)( 1 , 1 , … , 1 ). If follows from the proof of Lemma 1 that det𝐌(x,𝐚)=1𝐌𝑥𝐚1\det{\bf M}(x,{\bf a})=1roman_det bold_M ( italic_x , bold_a ) = 1. Further all entries of 𝐌(x0,𝐚0)𝐌subscript𝑥0subscript𝐚0{\bf M}(x_{0},{\bf a}_{0})bold_M ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) can be represented as polynomials in 1/e1𝑒1/e1 / italic_e. Thus, 𝐌(x0,𝐚0)1𝐌superscriptsubscript𝑥0subscript𝐚01{\bf M}(x_{0},{\bf a}_{0})^{-1}bold_M ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT has the same property and consequently 𝐚u(1)subscript𝐚𝑢1{\bf a}_{u}(1)bold_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( 1 ) has this property, too.

From that it directly follows from (31) that xuusubscript𝑥𝑢𝑢x_{uu}italic_x start_POSTSUBSCRIPT italic_u italic_u end_POSTSUBSCRIPT is also represented as a polynomial in 1/e1𝑒1/e1 / italic_e. (By definition, b(x,𝐚,u)𝑏𝑥𝐚𝑢b(x,{\bf a},u)italic_b ( italic_x , bold_a , italic_u ) is a rational polynomial of the entries of 𝐈𝐅𝐚𝐈subscript𝐅𝐚{\bf I}-{\bf F}_{\bf a}bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT.)

With help of (23) this finally leads to a representaion of σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT as a polynomial in 1/e1𝑒1/e1 / italic_e. ∎

This finally completes the proof of Theorem 1.

5. Extensions and Generalizations

In what follows we list some obvious and some less obvious extensions of our main result. For the sake of conciseness we do not present the details.

5.1. Several Patterns

Let {\mathcal{M}}caligraphic_M11{}_{1}start_FLOATSUBSCRIPT 1 end_FLOATSUBSCRIPT, \ldots, {\mathcal{M}}caligraphic_Mk𝑘{}_{k}start_FLOATSUBSCRIPT italic_k end_FLOATSUBSCRIPT be k𝑘kitalic_k different patterns. Then the problem is to determine the joint (limiting) distribution of the number of occurrences of {\mathcal{M}}caligraphic_M11{}_{1}start_FLOATSUBSCRIPT 1 end_FLOATSUBSCRIPT, \ldots, {\mathcal{M}}caligraphic_Mk𝑘{}_{k}start_FLOATSUBSCRIPT italic_k end_FLOATSUBSCRIPT in trees of size n𝑛nitalic_n. Using the same techniques as above (introducing the forest of planted patterns deduced from the patterns) we again obtain a system of functional equations. The only difference is that we now have to count occurrences of {\mathcal{M}}caligraphic_M11{}_{1}start_FLOATSUBSCRIPT 1 end_FLOATSUBSCRIPT, \ldots, {\mathcal{M}}caligraphic_Mk𝑘{}_{k}start_FLOATSUBSCRIPT italic_k end_FLOATSUBSCRIPT with different variables u1,,uksubscript𝑢1subscript𝑢𝑘u_{1},\ldots,u_{k}italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, which is done in the same fashion as for a single u𝑢uitalic_u. In view of Theorem 2, multiple variables u𝑢uitalic_u make no difference and we obtain a multivariate Gaussian limiting distribution.

5.2. Patterns Containing Paths of Unspecified Length

It might also be interesting to consider patterns where specific edges can be replaced by paths of arbitrary length. It turns out that this case in particular is more involved since a natural partition of all planted rooted trees is now infinite. Nevertheless it is possible to replace infinite series of such classes by one new class and end up with a finite system. Thus, this leads to a Gaussian limit law (as above).

5.3. Filled and Empty Nodes

In our model we have distinguished between internal (filled) and external (empty) nodes of the pattern {\mathcal{M}}caligraphic_M, where the degrees of the internal (filled) nodes have to match exactly. It also seems to be possible to consider the following more general matching problem: Let {\mathcal{M}}caligraphic_M again be a finite tree, where certain nodes are “filled” and the remaining ones are “empty”. Now we say that {\mathcal{M}}caligraphic_M matches if it occurs as a subtree such that the corresponding degrees of the filled nodes are equal whereas the degrees of the empty nodes might be different. It seems that the counting procedure above can be adapted to cover this case, too. However, it is definitely more involved. For example, if leaves of the pattern are filled nodes then these nodes have to be leaves wherever the pattern occurs. This implies that some of the functions aj(x,u)subscript𝑎𝑗𝑥𝑢a_{j}(x,u)italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , italic_u ) are then explicitly given in the system and the dependency graph is not strongly connected. However, it seems that this situation can be managed by eliminating these functions. Furthermore, and this is more serious, in general one has to consider infinitely many classes of trees leading to an infinite system of functional equations, in particular if an internal node is “empty”. In such a case Theorem 2 cannot be applied any more. Nevertheless we hope that the approach of Lalley [Lal], that is applicable to infinite systems of functional equations in one variable, can be generalized to a corresponding generalization of Theorem 2 to proper infinite systems. Thus, we can expect a Gaussian limit law even in this case.

In order to be more precise we will present an easy example. Let {\mathcal{M}}{}caligraphic_M denote the pattern depicted in Figure 7. Here all nodes are empty. Thus, the corresponding pattern counting problem is a subgraph counting problem.

Refer to caption
Figure 7. Example pattern with empty nodes

We partition all planted trees according to their root degree. Let aksubscript𝑎𝑘a_{k}italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT denote the set of planted rooted trees with root out-degree k𝑘kitalic_k and ak(x,u)subscript𝑎𝑘𝑥𝑢a_{k}(x,u)italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_x , italic_u ) the correponding generating function (that also counts the number of subgraph occcurences of {\mathcal{M}}{}caligraphic_M). Further, let r(x,u)𝑟𝑥𝑢r(x,u)italic_r ( italic_x , italic_u ) denote the generating function of rooted trees. Then we have

ak(x,u)=xk!(i0ai(x,u)u(k2)(i3)+(k3)(i2))k(k0)subscript𝑎𝑘𝑥𝑢𝑥𝑘superscriptsubscript𝑖0subscript𝑎𝑖𝑥𝑢superscript𝑢binomial𝑘2binomial𝑖3binomial𝑘3binomial𝑖2𝑘𝑘0a_{k}(x,u)=\frac{x}{k!}\left(\sum_{i\geq 0}a_{i}(x,u)u^{\binom{k}{2}\binom{i}{3}+\binom{k}{3}\binom{i}{2}}\right)^{k}\qquad(k\geq 0)italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_x , italic_u ) = divide start_ARG italic_x end_ARG start_ARG italic_k ! end_ARG ( ∑ start_POSTSUBSCRIPT italic_i ≥ 0 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_u ) italic_u start_POSTSUPERSCRIPT ( FRACOP start_ARG italic_k end_ARG start_ARG 2 end_ARG ) ( FRACOP start_ARG italic_i end_ARG start_ARG 3 end_ARG ) + ( FRACOP start_ARG italic_k end_ARG start_ARG 3 end_ARG ) ( FRACOP start_ARG italic_i end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_k ≥ 0 )

and

r(x,u)=xk01k!(i0ai(x,u)u(k12)(i3)+(k13)(i2))k.𝑟𝑥𝑢𝑥subscript𝑘01𝑘superscriptsubscript𝑖0subscript𝑎𝑖𝑥𝑢superscript𝑢binomial𝑘12binomial𝑖3binomial𝑘13binomial𝑖2𝑘r(x,u)=x\sum_{k\geq 0}\frac{1}{k!}\left(\sum_{i\geq 0}a_{i}(x,u)u^{\binom{k-1}{2}\binom{i}{3}+\binom{k-1}{3}\binom{i}{2}}\right)^{k}.italic_r ( italic_x , italic_u ) = italic_x ∑ start_POSTSUBSCRIPT italic_k ≥ 0 end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_k ! end_ARG ( ∑ start_POSTSUBSCRIPT italic_i ≥ 0 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_u ) italic_u start_POSTSUPERSCRIPT ( FRACOP start_ARG italic_k - 1 end_ARG start_ARG 2 end_ARG ) ( FRACOP start_ARG italic_i end_ARG start_ARG 3 end_ARG ) + ( FRACOP start_ARG italic_k - 1 end_ARG start_ARG 3 end_ARG ) ( FRACOP start_ARG italic_i end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT .

This system is easy to solve for u=1𝑢1u=1italic_u = 1. Here we have ak(x,1)=xp(x)k/k!subscript𝑎𝑘𝑥1𝑥𝑝superscript𝑥𝑘𝑘a_{k}(x,1)=xp(x)^{k}/k!italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_x italic_p ( italic_x ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT / italic_k ! and r(x,1)=p(x)𝑟𝑥1𝑝𝑥r(x,1)=p(x)italic_r ( italic_x , 1 ) = italic_p ( italic_x ). By taking derivatives with respect to u𝑢uitalic_u and summing over all k𝑘kitalic_k we also get (after some algebra)

ru(x,1)=512p(x)71p(x)+16p(x)81p(x)+p(x)76.subscript𝑟𝑢𝑥1512𝑝superscript𝑥71𝑝𝑥16𝑝superscript𝑥81𝑝𝑥𝑝superscript𝑥76r_{u}(x,1)=\frac{5}{12}\frac{p(x)^{7}}{1-p(x)}+\frac{1}{6}\frac{p(x)^{8}}{1-p(x)}+\frac{p(x)^{7}}{6}.italic_r start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_x , 1 ) = divide start_ARG 5 end_ARG start_ARG 12 end_ARG divide start_ARG italic_p ( italic_x ) start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_p ( italic_x ) end_ARG + divide start_ARG 1 end_ARG start_ARG 6 end_ARG divide start_ARG italic_p ( italic_x ) start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_p ( italic_x ) end_ARG + divide start_ARG italic_p ( italic_x ) start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT end_ARG start_ARG 6 end_ARG .

This implies that the average value of pattern occurences (in this sense) is of the form (7/12)n+O(1)712𝑛𝑂1(7/12)n+O(1)( 7 / 12 ) italic_n + italic_O ( 1 ), that is, μ=7/12𝜇712\mu=7/12italic_μ = 7 / 12. In principle it is also possible to get asymptotics for higher moments but the calculations get more and more involved.

5.4. Simply Generated Trees

Simply generated trees have been introduced by Meir and Moon [MM78] and are proper generalizations of several types of rooted trees. Let

φ(x)=φ0+φ1x+φ2x2+𝜑𝑥subscript𝜑0subscript𝜑1𝑥subscript𝜑2superscript𝑥2\varphi(x)=\varphi_{0}+\varphi_{1}x+\varphi_{2}x^{2}+\cdotsitalic_φ ( italic_x ) = italic_φ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x + italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ⋯

be a power series with non-negative coefficients; in particular we assume that φ0>0subscript𝜑00\varphi_{0}>0italic_φ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 and φj>0subscript𝜑𝑗0\varphi_{j}>0italic_φ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT > 0 for some j2𝑗2j\geq 2italic_j ≥ 2. We then define the weight ω(T)𝜔𝑇\omega(T)italic_ω ( italic_T ) of a finite rooted tree T𝑇Titalic_T by

ω(T)=j0φjDj(T),𝜔𝑇subscriptproduct𝑗0superscriptsubscript𝜑𝑗subscript𝐷𝑗𝑇\omega(T)=\prod_{j\geq 0}\varphi_{j}^{D_{j}(T)},italic_ω ( italic_T ) = ∏ start_POSTSUBSCRIPT italic_j ≥ 0 end_POSTSUBSCRIPT italic_φ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_T ) end_POSTSUPERSCRIPT ,

where Dj(T)subscript𝐷𝑗𝑇D_{j}(T)italic_D start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_T ) denotes the number of nodes in T𝑇Titalic_T with j𝑗jitalic_j successors. If we set

yn=|T|=nω(T)subscript𝑦𝑛subscript𝑇𝑛𝜔𝑇y_{n}=\sum_{|T|=n}\omega(T)italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT | italic_T | = italic_n end_POSTSUBSCRIPT italic_ω ( italic_T )

then the generating function

y(x)=n1ynxn𝑦𝑥subscript𝑛1subscript𝑦𝑛superscript𝑥𝑛y(x)=\sum_{n\geq 1}y_{n}x^{n}italic_y ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n ≥ 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT

satisfies the functional equation

y(x)=xφ(y(x)).𝑦𝑥𝑥𝜑𝑦𝑥y(x)=x\varphi(y(x)).italic_y ( italic_x ) = italic_x italic_φ ( italic_y ( italic_x ) ) .

In this context, ynsubscript𝑦𝑛y_{n}italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT denotes a weighted number of trees of size n𝑛nitalic_n. For example, if φj=1subscript𝜑𝑗1\varphi_{j}=1italic_φ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1 for all j0𝑗0j\geq 0italic_j ≥ 0 (that is, φ(x)=1/(1x)𝜑𝑥11𝑥\varphi(x)=1/(1-x)italic_φ ( italic_x ) = 1 / ( 1 - italic_x )) then all rooted trees have weight ω(T)=1𝜔𝑇1\omega(T)=1italic_ω ( italic_T ) = 1 and yn=pnsubscript𝑦𝑛subscript𝑝𝑛y_{n}=p_{n}italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is the number of planted plane trees. If φj=1/j!subscript𝜑𝑗1𝑗\varphi_{j}=1/j!italic_φ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1 / italic_j ! (that is, φ(x)=ex𝜑𝑥superscript𝑒𝑥\varphi(x)=e^{x}italic_φ ( italic_x ) = italic_e start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT) then we formally get labeled rooted trees, etc.

Of course, we can proceed in the same way as above and obtain a system of functional equations that counts occurrences of a specific pattern in simply generated trees, and (under suitable conditions on the growth of φjsubscript𝜑𝑗\varphi_{j}italic_φ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT) we finally obtain a Gaussian limiting distribution. This has explicitly been done by Kok in his thesis [Kok05a, Kok05b].

5.5. Unlabeled Trees

Let p^nsubscript^𝑝𝑛\hat{p}_{n}over^ start_ARG italic_p end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT denote the number of unlabeled planted rooted trees and t^nsubscript^𝑡𝑛\hat{t}_{n}over^ start_ARG italic_t end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT the number of unlabeled unrooted trees. The generating functions are denoted by

p^(x)=n1p^nxnandt^(x)=n1t^nxn.formulae-sequence^𝑝𝑥subscript𝑛1subscript^𝑝𝑛superscript𝑥𝑛and^𝑡𝑥subscript𝑛1subscript^𝑡𝑛superscript𝑥𝑛\hat{p}(x)=\sum_{n\geq 1}\hat{p}_{n}x^{n}\quad\mbox{and}\quad\hat{t}(x)=\sum_{n\geq 1}\hat{t}_{n}x^{n}.over^ start_ARG italic_p end_ARG ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n ≥ 1 end_POSTSUBSCRIPT over^ start_ARG italic_p end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and over^ start_ARG italic_t end_ARG ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n ≥ 1 end_POSTSUBSCRIPT over^ start_ARG italic_t end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .

The structure of these trees is much more difficult than that of labeled trees. It turns out that one has to apply Pólya’s theory of counting and an amazing observation (15) by Otter [Ott48]. The generating functions p^(x)^𝑝𝑥\hat{p}(x)over^ start_ARG italic_p end_ARG ( italic_x ) and t^(x)^𝑡𝑥\hat{t}(x)over^ start_ARG italic_t end_ARG ( italic_x ) satisfy the functional equations

p^(x)=xk0Z(Sk;p^(x),p^(x2),,p^(xk))=xexp(p^(x)+12p^(x2)+13p^(x3)+)^𝑝𝑥𝑥subscript𝑘0𝑍subscript𝑆𝑘^𝑝𝑥^𝑝superscript𝑥2^𝑝superscript𝑥𝑘𝑥^𝑝𝑥12^𝑝superscript𝑥213^𝑝superscript𝑥3\hat{p}(x)=x\sum_{k\geq 0}Z\left(S_{k};\hat{p}(x),\hat{p}(x^{2}),\ldots,\hat{p}(x^{k})\right)\\ =x\exp\left(\hat{p}(x)+\frac{1}{2}\hat{p}(x^{2})+\frac{1}{3}\hat{p}(x^{3})+\cdots\right)over^ start_ARG italic_p end_ARG ( italic_x ) = italic_x ∑ start_POSTSUBSCRIPT italic_k ≥ 0 end_POSTSUBSCRIPT italic_Z ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ; over^ start_ARG italic_p end_ARG ( italic_x ) , over^ start_ARG italic_p end_ARG ( italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , … , over^ start_ARG italic_p end_ARG ( italic_x start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) ) = italic_x roman_exp ( over^ start_ARG italic_p end_ARG ( italic_x ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG over^ start_ARG italic_p end_ARG ( italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 3 end_ARG over^ start_ARG italic_p end_ARG ( italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) + ⋯ )

and

(15) t^(x)=p^(x)12p^(x)2+12p^(x2),^𝑡𝑥^𝑝𝑥12^𝑝superscript𝑥212^𝑝superscript𝑥2\hat{t}(x)=\hat{p}(x)-\frac{1}{2}\hat{p}(x)^{2}+\frac{1}{2}\hat{p}(x^{2}),over^ start_ARG italic_t end_ARG ( italic_x ) = over^ start_ARG italic_p end_ARG ( italic_x ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG over^ start_ARG italic_p end_ARG ( italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG over^ start_ARG italic_p end_ARG ( italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ,

where Z(Sk;x1,,xk)𝑍subscript𝑆𝑘subscript𝑥1subscript𝑥𝑘Z(S_{k};x_{1},\ldots,x_{k})italic_Z ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ; italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) denotes the cycle index of the symmetric group Sksubscript𝑆𝑘S_{k}italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. These functions have a common radius of convergence ρ0.338219𝜌0.338219\rho\approx 0.338219italic_ρ ≈ 0.338219 and a local expansion of the form

p^(x)=1b(ρx)1/2+c(ρx)+d(ρx)3/2+𝒪((ρx)2))\hat{p}(x)=1-b(\rho-x)^{1/2}+c(\rho-x)+d(\rho-x)^{3/2}+{\mathcal{O}}\left((\rho-x)^{2})\right)over^ start_ARG italic_p end_ARG ( italic_x ) = 1 - italic_b ( italic_ρ - italic_x ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT + italic_c ( italic_ρ - italic_x ) + italic_d ( italic_ρ - italic_x ) start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT + caligraphic_O ( ( italic_ρ - italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) )

and

t^(x)=1+p^(ρ2)2b2+2ρp^(ρ2)2(ρx)+bc(ρx)3/2+𝒪((ρx)2)),\hat{t}(x)=\frac{1+\hat{p}(\rho^{2})}{2}-\frac{b^{2}+2\rho\hat{p}^{\prime}(\rho^{2})}{2}(\rho-x)+bc(\rho-x)^{3/2}+{\mathcal{O}}\left((\rho-x)^{2})\right),over^ start_ARG italic_t end_ARG ( italic_x ) = divide start_ARG 1 + over^ start_ARG italic_p end_ARG ( italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 end_ARG - divide start_ARG italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_ρ over^ start_ARG italic_p end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 end_ARG ( italic_ρ - italic_x ) + italic_b italic_c ( italic_ρ - italic_x ) start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT + caligraphic_O ( ( italic_ρ - italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ,

where b2.6811266𝑏2.6811266b\approx 2.6811266italic_b ≈ 2.6811266 and c=b2/32.3961466𝑐superscript𝑏232.3961466c=b^{2}/3\approx 2.3961466italic_c = italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 3 ≈ 2.3961466, and x=ρ𝑥𝜌x=\rhoitalic_x = italic_ρ is the only singularity on the circle of convergence |x|=ρ𝑥𝜌|x|=\rho| italic_x | = italic_ρ. Thus, they behave similarly as p(x)𝑝𝑥p(x)italic_p ( italic_x ) and t(x)𝑡𝑥t(x)italic_t ( italic_x ). We also get

p^n=bρ2πn3/2ρn(1+𝒪(n1))subscript^𝑝𝑛𝑏𝜌2𝜋superscript𝑛32superscript𝜌𝑛1𝒪superscript𝑛1\hat{p}_{n}=\frac{b\sqrt{\rho}}{2\sqrt{\pi}}n^{-3/2}\rho^{-n}\left(1+{\mathcal{O}}\left(n^{-1}\right)\right)over^ start_ARG italic_p end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = divide start_ARG italic_b square-root start_ARG italic_ρ end_ARG end_ARG start_ARG 2 square-root start_ARG italic_π end_ARG end_ARG italic_n start_POSTSUPERSCRIPT - 3 / 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ( 1 + caligraphic_O ( italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) )

and

t^n=b3ρ3/24πn5/2ρn(1+𝒪(n1)).subscript^𝑡𝑛superscript𝑏3superscript𝜌324𝜋superscript𝑛52superscript𝜌𝑛1𝒪superscript𝑛1\hat{t}_{n}=\frac{b^{3}\rho^{3/2}}{4\sqrt{\pi}}n^{-5/2}\rho^{-n}\left(1+{\mathcal{O}}\left(n^{-1}\right)\right).over^ start_ARG italic_t end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = divide start_ARG italic_b start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 square-root start_ARG italic_π end_ARG end_ARG italic_n start_POSTSUPERSCRIPT - 5 / 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ( 1 + caligraphic_O ( italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ) .

Furthermore, it is possible to count the number of nodes of specific degree with the help of bivariate generating functions (compare with [DG99]). Thus, using Pólya’s theory of counting we can also obtain a system of functional equations for bivariate generating functions that count the number of occurrences of a specific pattern. The major difference to the procedure above is that this system also contains terms of the form aj(xk,uk)subscript𝑎𝑗superscript𝑥𝑘superscript𝑢𝑘a_{j}(x^{k},u^{k})italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_u start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) for k2𝑘2k\geq 2italic_k ≥ 2. Fortunately these terms can be considered as known functions when x𝑥xitalic_x varies around the singularity ρ𝜌\rhoitalic_ρ and u𝑢uitalic_u varies around 1111 (compare again with [DG99]). Hence, Theorem 2 applies again and we can proceed as above. This has explicitly been done by Kok in his thesis [Kok05a, Kok05b].

5.6. Forests

First, let us consider the case of labeled trees with generating function t(x,u)𝑡𝑥𝑢t(x,u)italic_t ( italic_x , italic_u ). Then the generating function f(x,u)𝑓𝑥𝑢f(x,u)italic_f ( italic_x , italic_u ) of unlabeled forests is given by

f(x,u)=et(x,u).𝑓𝑥𝑢superscript𝑒𝑡𝑥𝑢f(x,u)=e^{t(x,u)}.italic_f ( italic_x , italic_u ) = italic_e start_POSTSUPERSCRIPT italic_t ( italic_x , italic_u ) end_POSTSUPERSCRIPT .

Thus, the singular behaviour of f(x,u)𝑓𝑥𝑢f(x,u)italic_f ( italic_x , italic_u ) is the same as that of t(x,u)𝑡𝑥𝑢t(x,u)italic_t ( italic_x , italic_u ) (compare with [DG99]) and consequently we again obtain a Gaussian limiting distribution for the number of occurrences of a specific pattern in labeled forests.

The case of unlabeled forests is similar. Here we have

f^(x,u)=exp(t^(x,u)+12t^(x2,u2)+13t^(x3,u3)+).^𝑓𝑥𝑢^𝑡𝑥𝑢12^𝑡superscript𝑥2superscript𝑢213^𝑡superscript𝑥3superscript𝑢3\hat{f}(x,u)=\exp\left(\hat{t}(x,u)+\frac{1}{2}\hat{t}(x^{2},u^{2})+\frac{1}{3}\hat{t}(x^{3},u^{3})+\cdots\right).over^ start_ARG italic_f end_ARG ( italic_x , italic_u ) = roman_exp ( over^ start_ARG italic_t end_ARG ( italic_x , italic_u ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG over^ start_ARG italic_t end_ARG ( italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 3 end_ARG over^ start_ARG italic_t end_ARG ( italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) + ⋯ ) .

Of course, we can consider other classes of trees or forests of a given number of trees.

5.7. Forbidden Patterns

It is also interesting to count the number tn,0subscript𝑡𝑛0t_{n,0}italic_t start_POSTSUBSCRIPT italic_n , 0 end_POSTSUBSCRIPT of trees of size n𝑛nitalic_n without a given pattern. The generating function of these numbers is just p(x,0)𝑝𝑥0p(x,0)italic_p ( italic_x , 0 ), resp. t(x,0)𝑡𝑥0t(x,0)italic_t ( italic_x , 0 ). It is now an easy exercise to show that there exists an η>0𝜂0\eta>0italic_η > 0 such that

tn,0tneηn.subscript𝑡𝑛0subscript𝑡𝑛superscript𝑒𝜂𝑛t_{n,0}\leq t_{n}e^{-\eta n}.italic_t start_POSTSUBSCRIPT italic_n , 0 end_POSTSUBSCRIPT ≤ italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_η italic_n end_POSTSUPERSCRIPT .

The only thing we have to check is that the radius of convergence of t(x,0)𝑡𝑥0t(x,0)italic_t ( italic_x , 0 ) is larger than the radius of convergence of t(x,1)𝑡𝑥1t(x,1)italic_t ( italic_x , 1 ). However, this is obvious since the radius of convergence of t(x,u)𝑡𝑥𝑢t(x,u)italic_t ( italic_x , italic_u ) (which is the same as that of p(x,u)𝑝𝑥𝑢p(x,u)italic_p ( italic_x , italic_u )) is given by x(u)𝑥𝑢x(u)italic_x ( italic_u ) (for u𝑢uitalic_u around 1111) and x(1)<0superscript𝑥10x^{\prime}(1)<0italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 1 ) < 0.

Appendix A Algorithms

In the main part of this paper we showed that the limiting distribution of the number of pattern occurrences is normal with computable μ𝜇\muitalic_μ and σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. However the family of classes {a0,a1,,aL}subscript𝑎0subscript𝑎1subscript𝑎𝐿\{a_{0},a_{1},\dots,a_{L}\}{ italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT } considered in the first part was especially created to make the arguments more transparent, there were no considerations about minimality. In this appendix we focus on creating another partition 𝒜={a0,,aL}𝒜subscript𝑎0subscript𝑎𝐿\mathcal{A}=\{a_{0},\dots,a_{L}\}caligraphic_A = { italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT } of p𝑝pitalic_p which has considerably less classes. It also has the properties that it is recursively describable and allows an unambiguous definition of the number of additional occurrences K(l0,,lL)𝐾subscript𝑙0subscript𝑙𝐿K(l_{0},\dots,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) of the pattern. For example we show that for the pattern of Figure 9 we need just 8 equations whereas the previous proof would use more than 1000 equations.

First we remark that in some cases it is profitable to adjust the structure of the system of equations (1) in Proposition 1 by allowing an additional polynomial P0(y0,,yL,u)subscript𝑃0subscript𝑦0subscript𝑦𝐿𝑢P_{0}(y_{0},\dots,y_{L},u)italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_u ) in the first equation. The first equation then becomes

a0(x,u)subscript𝑎0𝑥𝑢\displaystyle a_{0}(x,u)italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) =xP0(a0(x,u),,aL(x,u),u)absent𝑥subscript𝑃0subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢𝑢\displaystyle=x\cdot P_{0}(a_{0}(x,u),\ldots,a_{L}(x,u),u)= italic_x ⋅ italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) , italic_u )
+(xea0(x,u)++aL(x,u)xj=0LPj(a0(x,u),,aL(x,u),1)).𝑥superscript𝑒subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢𝑥superscriptsubscript𝑗0𝐿subscript𝑃𝑗subscript𝑎0𝑥𝑢subscript𝑎𝐿𝑥𝑢1\displaystyle\qquad+(xe^{a_{0}(x,u)+\cdots+a_{L}(x,u)}-x\sum_{j=0}^{L}P_{j}(a_{0}(x,u),\ldots,a_{L}(x,u),1)).+ ( italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) end_POSTSUPERSCRIPT - italic_x ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , italic_u ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , italic_u ) , 1 ) ) .

This system still fits our analytical framework. The advantage is that for example the minimal system of equations for counting stars in trees on page 1 now fits this modified system.

The idea for constructing 𝒜𝒜\mathcal{A}caligraphic_A will be to create in a first time a certain family of tree classes 𝒮={t1,,tn}𝒮subscript𝑡1subscript𝑡𝑛\mathcal{S}=\{t_{1},\dots,t_{n}\}caligraphic_S = { italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }, not necessarily building a partition of p𝑝pitalic_p. Each of these classes will be defined as the class of all trees in p𝑝pitalic_p which “start” in a certain way, or with other words, which match a certain tree tisuperscriptsubscript𝑡𝑖t_{i}^{\prime}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT at the root, just as was the case for the aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in the main part of this paper. By abuse of notation we will usually write tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT instead of tisuperscriptsubscript𝑡𝑖t_{i}^{\prime}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for this tree. Let J={1,,n}𝐽1𝑛J=\{1,\dots,n\}italic_J = { 1 , … , italic_n } and tic=ptisuperscriptsubscript𝑡𝑖𝑐𝑝subscript𝑡𝑖t_{i}^{c}=p\setminus t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = italic_p ∖ italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Now, by collecting in 𝒜𝒜\mathcal{A}caligraphic_A all different, non-empty classes of the form

(16) aI=iItiiJItic,IJformulae-sequencesubscript𝑎𝐼subscript𝑖𝐼subscript𝑡𝑖subscript𝑖𝐽𝐼superscriptsubscript𝑡𝑖𝑐𝐼𝐽\displaystyle a_{I}=\bigcap_{i\in I}t_{i}\cap\bigcap_{i\in J\setminus I}t_{i}^{c},\qquad I\subseteq Jitalic_a start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT = ⋂ start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ ⋂ start_POSTSUBSCRIPT italic_i ∈ italic_J ∖ italic_I end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT , italic_I ⊆ italic_J

we will obtain a partition 𝒜𝒜\mathcal{A}caligraphic_A of p𝑝pitalic_p. This partition will have a recursive description by construction, see the algorithms below. Furthermore, if 𝒮𝒮\mathcal{S}caligraphic_S is sufficiently rich, this partition will allow an unambiguous definition of K(l0,,lL)𝐾subscript𝑙0subscript𝑙𝐿K(l_{0},\dots,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ).

We now make some considerations about the properties that 𝒮𝒮\mathcal{S}caligraphic_S should possess to make sure that 𝒜𝒜\mathcal{A}caligraphic_A will allow an unambiguous definition of K(l0,,lL)𝐾subscript𝑙0subscript𝑙𝐿K(l_{0},\dots,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ). Let b𝑏bitalic_b be a subclass of p𝑝pitalic_p. For each tree Tp𝑇𝑝T\in pitalic_T ∈ italic_p we can determine the number k(T)𝑘𝑇k(T)italic_k ( italic_T ) of pattern occurrences at the root of T𝑇Titalic_T. Let k(b)={k(T):Tb}𝑘𝑏conditional-set𝑘𝑇𝑇𝑏k({b})=\{\,k(T):T\in b\,\}italic_k ( italic_b ) = { italic_k ( italic_T ) : italic_T ∈ italic_b }. Because the patterns have finitely many nodes and because in each internal node the degree is fixed and the root has to be part of the match, there are only finitely many ways for a pattern match. Thus the set k(b)𝑘𝑏k({b})italic_k ( italic_b ) will be finite and non-empty. Now let aIsubscript𝑎𝐼a_{I}italic_a start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT defined by equation (16) (and non-empty). Now it holds that

(17) k(aI)iIk(ti)iJIk(tic)𝑘subscript𝑎𝐼subscript𝑖𝐼𝑘subscript𝑡𝑖subscript𝑖𝐽𝐼𝑘superscriptsubscript𝑡𝑖𝑐\displaystyle k(a_{I})\subseteq\bigcap_{i\in I}k(t_{i})\cap\bigcap_{i\in J\setminus I}k(t_{i}^{c})italic_k ( italic_a start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ) ⊆ ⋂ start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT italic_k ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∩ ⋂ start_POSTSUBSCRIPT italic_i ∈ italic_J ∖ italic_I end_POSTSUBSCRIPT italic_k ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT )

because a tree T𝑇Titalic_T in aIsubscript𝑎𝐼a_{I}italic_a start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT is by definition in ti,iIsubscript𝑡𝑖𝑖𝐼t_{i},\ i\in Iitalic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_i ∈ italic_I and tic,iJIsuperscriptsubscript𝑡𝑖𝑐𝑖𝐽𝐼t_{i}^{c},\ i\in J\setminus Iitalic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT , italic_i ∈ italic_J ∖ italic_I, thus the number of pattern occurrences at the root is constrained by k(ti),iI𝑘subscript𝑡𝑖𝑖𝐼k(t_{i}),\ i\in Iitalic_k ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_i ∈ italic_I and k(tic),iJI𝑘superscriptsubscript𝑡𝑖𝑐𝑖𝐽𝐼k(t_{i}^{c}),\ i\in J\setminus Iitalic_k ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) , italic_i ∈ italic_J ∖ italic_I. If 𝒮={t1,,tn}𝒮subscript𝑡1subscript𝑡𝑛\mathcal{S}=\{t_{1},\dots,t_{n}\}caligraphic_S = { italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } is sufficiently rich, then k(aI)𝑘subscript𝑎𝐼k(a_{I})italic_k ( italic_a start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ) will only consist of a single number. This will be the case if for each m𝑚m\in\mathbb{N}italic_m ∈ blackboard_N, the family 𝒮𝒮\mathcal{S}caligraphic_S contains all classes of trees “starting” with all possible arrangements of m𝑚mitalic_m overlapping patterns. Indeed, if we have for example for a certain tree class tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT that k(ti)={r,r+1}𝑘subscript𝑡𝑖𝑟𝑟1k(t_{i})=\{r,r+1\}italic_k ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = { italic_r , italic_r + 1 }, then there will be another tree class tjsubscript𝑡𝑗t_{j}italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, which is a subclass of tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT with k(tj)={r+1}𝑘subscript𝑡𝑗𝑟1k(t_{j})=\{r+1\}italic_k ( italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = { italic_r + 1 }. Now the intersections b=titjc𝑏subscript𝑡𝑖superscriptsubscript𝑡𝑗𝑐b=t_{i}\cap t_{j}^{c}italic_b = italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT and c=titj𝑐subscript𝑡𝑖subscript𝑡𝑗c=t_{i}\cap t_{j}italic_c = italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT will yield tree classes with a singleton k(.)k(.)italic_k ( . ), namely k(b)={r}𝑘𝑏𝑟k(b)=\{r\}italic_k ( italic_b ) = { italic_r } and k(c)={r+1}𝑘𝑐𝑟1k(c)=\{r+1\}italic_k ( italic_c ) = { italic_r + 1 }.

For example consider a pattern which consists of a node of degree 2 attached to a node of degree 3. The corresponding planted patterns are shown in Figure 8. Now let 𝒮𝒮\mathcal{S}caligraphic_S consist of the three classes t1,t2,t3subscript𝑡1subscript𝑡2subscript𝑡3t_{1},t_{2},t_{3}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, shown in the center of Figure 8. We have k(t1)={1}𝑘subscript𝑡11k(t_{1})=\{1\}italic_k ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = { 1 }, because the left planted pattern surely matches and the other does not, k(t2)={1,2}𝑘subscript𝑡212k(t_{2})=\{1,2\}italic_k ( italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = { 1 , 2 }, because the left planted pattern does not match and the right one matches at least once, but possibly twice. k(t3)={2}𝑘subscript𝑡32k(t_{3})=\{2\}italic_k ( italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = { 2 }, because the left pattern does not match and the right one surely matches twice. We see that the only non-empty intersections of the form (16) are a=t1t2ct3c𝑎subscript𝑡1superscriptsubscript𝑡2𝑐superscriptsubscript𝑡3𝑐a=t_{1}\cap t_{2}^{c}\cap t_{3}^{c}italic_a = italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT, b=t1ct2t3c𝑏superscriptsubscript𝑡1𝑐subscript𝑡2superscriptsubscript𝑡3𝑐b=t_{1}^{c}\cap t_{2}\cap t_{3}^{c}italic_b = italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∩ italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT and c=t1ct2t3𝑐superscriptsubscript𝑡1𝑐subscript𝑡2subscript𝑡3c=t_{1}^{c}\cap t_{2}\cap t_{3}italic_c = italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∩ italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. We obtain k(a)=k(b)={1}𝑘𝑎𝑘𝑏1k(a)=k(b)=\{1\}italic_k ( italic_a ) = italic_k ( italic_b ) = { 1 } and k(c)={2}𝑘𝑐2k(c)=\{2\}italic_k ( italic_c ) = { 2 }, which are all singletons.

Refer to caption
Figure 8. On the left: Planted patterns. Center: Classes tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Right: Classes {a,b,c}𝑎𝑏𝑐\{a,b,c\}{ italic_a , italic_b , italic_c }. The white box here means a node of out-degree different from 1. Note: this does not correspond to the output of the algorithms of this appendix

Because we also need a recursive description of the final partition 𝒜𝒜\mathcal{A}caligraphic_A, we will construct some additional tree classes tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. As the partition becomes finer when dealing with more classes tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, it is clear that k𝑘kitalic_k remains well-defined.

On the other hand we do not have to associate a unique number to k(aI)𝑘subscript𝑎𝐼k(a_{I})italic_k ( italic_a start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ), only to K(l0,,lL)𝐾subscript𝑙0subscript𝑙𝐿K(l_{0},\dots,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ). Therefore we can slightly reduce the family 𝒮={t1,,tn}𝒮subscript𝑡1subscript𝑡𝑛\mathcal{S}=\{t_{1},\dots,t_{n}\}caligraphic_S = { italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }. In the algorithm below this reduction of 𝒮𝒮\mathcal{S}caligraphic_S corresponds to considering only proper subtrees of the trees q𝒬𝑞𝒬q\in\mathcal{Q}italic_q ∈ caligraphic_Q (q𝑞qitalic_q itself is excluded).

A coarse-grain description of an algorithm now follows.

  1. (1)

    Calculate the set 𝒰𝒰\mathcal{U}caligraphic_U of all planar embeddings of all planted patterns deducible from the pattern \mathcal{M}caligraphic_M.

  2. (2)

    Consider the planted planar trees issue of step 1 as planar tree classes and take all possible intersections of any number of those classes. Now take the implied non-planar general tree structure of each class and collect these non-planar planted trees in the set 𝒬𝒬\mathcal{Q}caligraphic_Q.

  3. (3)

    Create a family 𝒮={t1,,tn}𝒮subscript𝑡1subscript𝑡𝑛\mathcal{S}=\{t_{1},\dots,t_{n}\}caligraphic_S = { italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } for the forest of planted subtrees of trees q𝒬𝑞𝒬q\in\mathcal{Q}italic_q ∈ caligraphic_Q, excluding the trees q𝑞qitalic_q themselves, where each tjsubscript𝑡𝑗t_{j}italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT has a recursive description in t0,t1,,tj1subscript𝑡0subscript𝑡1subscript𝑡𝑗1t_{0},t_{1},\dots,t_{j-1}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT and where t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT denotes a leaf.

  4. (4)

    Now interpret t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT as the class of all trees p𝑝pitalic_p and interpret the trees ti𝒮subscript𝑡𝑖𝒮t_{i}\in\mathcal{S}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_S as non-planar tree classes. Construct a partition 𝒜={a0,,aL}𝒜subscript𝑎0subscript𝑎𝐿\mathcal{A}=\{a_{0},\dots,a_{L}\}caligraphic_A = { italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT } of the class of all planted trees p𝑝pitalic_p together with a recursive description (compare with (16)).

  5. (5)

    Calculate for each term in the recursive description the number K(l0,,lL)𝐾subscript𝑙0subscript𝑙𝐿K(l_{0},\dots,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) of additional pattern occurrences and deduce a system of equations for the generating functions aj(x,u)subscript𝑎𝑗𝑥𝑢a_{j}(x,u)italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , italic_u ) of the classes ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT.

Before giving more detailed algorithms, we give an example. Consider the pattern of Figure 9.

Refer to caption
Figure 9. Example pattern \mathcal{M}caligraphic_M

With the procedure of the main part of the article we would end up with more than 1000 classes, yielding a system of equations with the same number of equations. However, by using the following refined algorithm we only need 8 classes.

In the first step we create all planar embeddings of the corresponding planted pattern (trees τ1,τ2,τ3subscript𝜏1subscript𝜏2subscript𝜏3\tau_{1},\tau_{2},\tau_{3}italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT of Figure 14). This yields 32+2+42=1632242163\cdot 2+2+4\cdot 2=163 ⋅ 2 + 2 + 4 ⋅ 2 = 16 planar trees of which some are shown in Figure 10.

Refer to caption
Figure 10. Some of in total 16 planted planar embeddings 𝒰𝒰\mathcal{U}caligraphic_U

We now consider these structures as planar tree classes and additionally construct tree classes by taking all possible intersections of any number of the classes issued from step 1. Then, we take the non-planar implied tree structure of each planar class and collect these trees in 𝒬𝒬\mathcal{Q}caligraphic_Q. We end up with 24 different trees: 9 that stem from τ1subscript𝜏1\tau_{1}italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, 1 from τ2subscript𝜏2\tau_{2}italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, and 14 from τ3subscript𝜏3\tau_{3}italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. Some of them are shown in Figure 11.

Refer to caption
Figure 11. Some of in total 24 non-planar trees of 𝒬𝒬\mathcal{Q}caligraphic_Q

For all proper subtrees for each tree in 𝒬𝒬\mathcal{Q}caligraphic_Q we now construct a recursive description. For example, for the leftmost tree of Figure 11 we first consider the subtree consisting of a node with four leaves. We denote this class by t4=xt04subscript𝑡4𝑥superscriptsubscript𝑡04t_{4}=xt_{0}^{4}italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT. (Here we use the following structural notation: x𝑥xitalic_x denotes a root node, t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT a leaf and xt04𝑥superscriptsubscript𝑡04xt_{0}^{4}italic_x italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT denotes a root to which are attached 4 leaves.) The next subtree is a root of out-degree 2 to which a subtree of type t4subscript𝑡4t_{4}italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT is attached. We denote this with t5=xt0t4subscript𝑡5𝑥subscript𝑡0subscript𝑡4t_{5}=xt_{0}t_{4}italic_t start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. Figure 12 shows all 6 trees we end up with. Observe on our example that the collection of subtrees at the root extracted from the 24 trees in 𝒬𝒬\mathcal{Q}caligraphic_Q consists of only 6 trees.

Refer to caption
Figure 12. Non-planar trees tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT which possess a recursive description

Their recursive descriptions are given by

(18) t1=xt03,t2=xt0t1,t3=xt12,t4=xt04,t5=xt0t4,t6=xt42.formulae-sequencesubscript𝑡1𝑥superscriptsubscript𝑡03formulae-sequencesubscript𝑡2𝑥subscript𝑡0subscript𝑡1formulae-sequencesubscript𝑡3𝑥superscriptsubscript𝑡12formulae-sequencesubscript𝑡4𝑥superscriptsubscript𝑡04formulae-sequencesubscript𝑡5𝑥subscript𝑡0subscript𝑡4subscript𝑡6𝑥superscriptsubscript𝑡42\displaystyle t_{1}=xt_{0}^{3},\quad t_{2}=xt_{0}t_{1},\quad t_{3}=xt_{1}^{2},\quad t_{4}=xt_{0}^{4},\quad t_{5}=xt_{0}t_{4},\quad t_{6}=xt_{4}^{2}.italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_t start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

We now interpret t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT in (18) as the class of all planted trees p𝑝pitalic_p. The other tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are also interpreted as tree classes. For example, t1subscript𝑡1t_{1}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the class of all trees with root out-degree 3. We now construct a partition based on these classes and their recursive description of (18). We obtain the classes of Figure 13.

Refer to caption
Figure 13. Non-planar partition classes. The white box means “not out-degree 3 or 4” and the white triangle means “anything that is not contained in the other classes”

Their recursive description is given by

(19) a0subscript𝑎0\displaystyle a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =pi=17ai=xxi=07aix(a0a2a3a5a6a7)2xn=5(i=07ai)n,absent𝑝superscriptsubscriptdirect-sum𝑖17subscript𝑎𝑖direct-sum𝑥𝑥superscriptsubscriptdirect-sum𝑖07subscript𝑎𝑖𝑥superscriptdirect-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎72𝑥superscriptsubscriptdirect-sum𝑛5superscriptsuperscriptsubscriptdirect-sum𝑖07subscript𝑎𝑖𝑛\displaystyle=p\setminus\bigoplus_{i=1}^{7}a_{i}=x\oplus x\bigoplus_{i=0}^{7}a_{i}\oplus x(a_{0}\oplus a_{2}\oplus a_{3}\oplus a_{5}\oplus a_{6}\oplus a_{7})^{2}\oplus x\bigoplus_{n=5}^{\infty}\left(\bigoplus_{i=0}^{7}a_{i}\right)^{n},= italic_p ∖ ⨁ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x ⊕ italic_x ⨁ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⊕ italic_x ⨁ start_POSTSUBSCRIPT italic_n = 5 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( ⨁ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ,
a1subscript𝑎1\displaystyle a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =xp3,absent𝑥superscript𝑝3\displaystyle=xp^{3},= italic_x italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ,
a2subscript𝑎2\displaystyle a_{2}italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =xa12,absent𝑥superscriptsubscript𝑎12\displaystyle=xa_{1}^{2},= italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a3subscript𝑎3\displaystyle a_{3}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT =xa1a4,absent𝑥subscript𝑎1subscript𝑎4\displaystyle=xa_{1}a_{4},= italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ,
a4subscript𝑎4\displaystyle a_{4}italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT =xp4,absent𝑥superscript𝑝4\displaystyle=xp^{4},= italic_x italic_p start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ,
a5subscript𝑎5\displaystyle a_{5}italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT =x(a0a2a3a5a6a7)a1,absent𝑥direct-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎1\displaystyle=x(a_{0}\oplus a_{2}\oplus a_{3}\oplus a_{5}\oplus a_{6}\oplus a_{7})a_{1},= italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ,
a6subscript𝑎6\displaystyle a_{6}italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT =xa42,absent𝑥superscriptsubscript𝑎42\displaystyle=xa_{4}^{2},= italic_x italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a7subscript𝑎7\displaystyle a_{7}italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT =x(a0a2a3a5a6a7)a4.absent𝑥direct-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎4\displaystyle=x(a_{0}\oplus a_{2}\oplus a_{3}\oplus a_{5}\oplus a_{6}\oplus a_{7})a_{4}.= italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT .

The last step consists of determining the number of additional occurrences K(l0,,l7)𝐾subscript𝑙0subscript𝑙7K(l_{0},\dots,l_{7})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) for each term in the recursive description (19) and translating (19) in a system of equations for the generating functions aj(x,u)=ajsubscript𝑎𝑗𝑥𝑢subscript𝑎𝑗a_{j}(x,u)=a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , italic_u ) = italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT. As an example we consider the equation for a1subscript𝑎1a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Class a1subscript𝑎1a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT consists of the trees of root out-degree 3. We get no additional occurrences of the pattern if we attach a tree of class a0,a1,a2,a4subscript𝑎0subscript𝑎1subscript𝑎2subscript𝑎4a_{0},a_{1},a_{2},a_{4}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT or a5subscript𝑎5a_{5}italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT to such a root, we get one additional occurrence for each tree of class a3subscript𝑎3a_{3}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT or a7subscript𝑎7a_{7}italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT and we have two additional occurrences for each tree of class a6subscript𝑎6a_{6}italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT attached to the root. This yields the equation for a1(x,u)subscript𝑎1𝑥𝑢a_{1}(x,u)italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_u ) below. Altogether we obtain:

a0subscript𝑎0\displaystyle a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =x+xi=07ai+12x(a0+a2+a3+a5+a6+a7)2+xn51n!(i=07ai)n,absent𝑥𝑥superscriptsubscript𝑖07subscript𝑎𝑖12𝑥superscriptsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎72𝑥subscript𝑛51𝑛superscriptsuperscriptsubscript𝑖07subscript𝑎𝑖𝑛\displaystyle=x+x\sum_{i=0}^{7}a_{i}+\frac{1}{2}x(a_{0}+a_{2}+a_{3}+a_{5}+a_{6}+a_{7})^{2}+x\sum_{n\geq 5}\frac{1}{n!}\left(\sum_{i=0}^{7}a_{i}\right)^{n},= italic_x + italic_x ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_x ∑ start_POSTSUBSCRIPT italic_n ≥ 5 end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_n ! end_ARG ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ,
a1subscript𝑎1\displaystyle a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =13!x(a0+a1+a2+a4+a5+(a3+a7)u+a6u2)3,absent13𝑥superscriptsubscript𝑎0subscript𝑎1subscript𝑎2subscript𝑎4subscript𝑎5subscript𝑎3subscript𝑎7𝑢subscript𝑎6superscript𝑢23\displaystyle=\frac{1}{3!}x(a_{0}+a_{1}+a_{2}+a_{4}+a_{5}+(a_{3}+a_{7})u+a_{6}u^{2})^{3},= divide start_ARG 1 end_ARG start_ARG 3 ! end_ARG italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) italic_u + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ,
a2subscript𝑎2\displaystyle a_{2}italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =12xa12,absent12𝑥superscriptsubscript𝑎12\displaystyle=\frac{1}{2}xa_{1}^{2},= divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a3subscript𝑎3\displaystyle a_{3}italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT =xa1a4u,absent𝑥subscript𝑎1subscript𝑎4𝑢\displaystyle=xa_{1}a_{4}u,= italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT italic_u ,
a4subscript𝑎4\displaystyle a_{4}italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT =14!x(a0+a1+a4+a6+a7+(a3+a5)u+a6u2)4,absent14𝑥superscriptsubscript𝑎0subscript𝑎1subscript𝑎4subscript𝑎6subscript𝑎7subscript𝑎3subscript𝑎5𝑢subscript𝑎6superscript𝑢24\displaystyle=\frac{1}{4!}x(a_{0}+a_{1}+a_{4}+a_{6}+a_{7}+(a_{3}+a_{5})u+a_{6}u^{2})^{4},= divide start_ARG 1 end_ARG start_ARG 4 ! end_ARG italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT + ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ) italic_u + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ,
a5subscript𝑎5\displaystyle a_{5}italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT =x(a0+a2+a3+a5+a6+a7)a1,absent𝑥subscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎1\displaystyle=x(a_{0}+a_{2}+a_{3}+a_{5}+a_{6}+a_{7})a_{1},= italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ,
a6subscript𝑎6\displaystyle a_{6}italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT =12xa42,absent12𝑥superscriptsubscript𝑎42\displaystyle=\frac{1}{2}xa_{4}^{2},= divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
a7subscript𝑎7\displaystyle a_{7}italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT =x(a0+a2+a3+a5+a6+a7)a4.absent𝑥subscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎4\displaystyle=x(a_{0}+a_{2}+a_{3}+a_{5}+a_{6}+a_{7})a_{4}.= italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT .

We can now calculate μ𝜇\muitalic_μ. We get μ=25643e8e3=0.865759040𝜇25643𝑒8superscript𝑒30.865759040\mu=\frac{256-43e}{8e^{3}}=0.865759040\dotsitalic_μ = divide start_ARG 256 - 43 italic_e end_ARG start_ARG 8 italic_e start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG = 0.865759040 …. The computation of σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT was not feasible, because of memory problems.555The actual computation uses polynomial expressions with more than 200,000 terms. We used Maple 9.5, which used up the memory of 1 GB and a very large part of the 1 GB swap.

A.1. Planar embedding algorithm: GeneralToPlanar


Input: a general planted tree τ𝜏\tauitalic_τ

Output: the set 𝒰𝒰\mathcal{U}caligraphic_U of planted planar trees π𝜋\piitalic_π that share τ𝜏\tauitalic_τ as their implied general tree structure

Algorithm:

  1. (1)

    write τ𝜏\tauitalic_τ in the form xτ1τk𝑥subscript𝜏1subscript𝜏𝑘x\tau_{1}\dotsm\tau_{k}italic_x italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_τ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, that is, let k𝑘kitalic_k be the root out-degree of τ𝜏\tauitalic_τ and τ1subscript𝜏1\tau_{1}italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, …, τksubscript𝜏𝑘\tau_{k}italic_τ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT be the children at the root

  2. (2)

    for each i𝑖iitalic_i between 1 and k𝑘kitalic_k, recursively compute Pi=GeneralToPlanar(τi)subscript𝑃𝑖GeneralToPlanarsubscript𝜏𝑖P_{i}=\mathrm{GeneralToPlanar}(\tau_{i})italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_GeneralToPlanar ( italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )

  3. (3)

    construct and return the set of planar trees xπσ(1)πσ(k)𝑥subscript𝜋𝜎1subscript𝜋𝜎𝑘x\pi_{\sigma(1)}\dotsm\pi_{\sigma(k)}italic_x italic_π start_POSTSUBSCRIPT italic_σ ( 1 ) end_POSTSUBSCRIPT ⋯ italic_π start_POSTSUBSCRIPT italic_σ ( italic_k ) end_POSTSUBSCRIPT over all choices of πiPisubscript𝜋𝑖subscript𝑃𝑖\pi_{i}\in P_{i}italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and over all permutations σ𝜎\sigmaitalic_σ of {1,,k}1𝑘\{1,\dots,k\}{ 1 , … , italic_k }

A.2. Tree class intersection algorithm


Input: a set of planted planar trees 𝒰𝒰\mathcal{U}caligraphic_U

Output: the set 𝒬𝒬\mathcal{Q}caligraphic_Q of non-planar planted trees which are obtained by intersecting planar tree classes based on 𝒰𝒰\mathcal{U}caligraphic_U and collecting the non-planar tree structures of the resulting planar tree classes.

Algorithm:

  1. (1)

    For each i𝑖iitalic_i between 1 and |𝒰|𝒰|\mathcal{U}|| caligraphic_U |, consider all i𝑖iitalic_i-tuples of different trees π1,,πi𝒰subscript𝜋1subscript𝜋𝑖𝒰\pi_{1},\dots,\pi_{i}\in\mathcal{U}italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_U and determine for each i𝑖iitalic_i-tuple if s=πiπi𝑠subscript𝜋𝑖subscript𝜋𝑖s=\pi_{i}\cap\dots\cap\pi_{i}italic_s = italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ ⋯ ∩ italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT may be interpreted as a non-empty tree class. In that case, let ssuperscript𝑠s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be the implied non-planar tree structure of s𝑠sitalic_s and add ssuperscript𝑠s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT to the set 𝒬𝒬\mathcal{Q}caligraphic_Q.

A.3. DAGification algorithm

  
We construct a recursive description for the forest of planted subtrees for each tree in a given set of planted trees. Here we do not consider the tree itself as a subtree of itself. This calculation is reminiscent of the DAGification process of computer science (see, e.g., [ASU86]), which aims at compacting an expression tree by sharing repeated subexpressions. However, if we interpret those subtrees as classes, the intersection of two classes need not be empty.


Input: set of planted trees 𝒬𝒬\mathcal{Q}caligraphic_Q


Output: a number m𝑚mitalic_m and a recursive description of the forest of planted subtrees 𝒮={t1,,tm}𝒮subscript𝑡1subscript𝑡𝑚\mathcal{S}=\{t_{1},\dots,t_{m}\}caligraphic_S = { italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } of the trees of 𝒬𝒬\mathcal{Q}caligraphic_Q, of the form

ti=xtλ1(i)tλri(i)(ri)for 1imsubscript𝑡𝑖𝑥subscript𝑡superscriptsubscript𝜆1𝑖subscript𝑡subscriptsuperscript𝜆𝑖subscript𝑟𝑖subscript𝑟𝑖for 1imt_{i}=xt_{\lambda_{1}^{(i)}}\dotsm t_{\lambda^{(i)}_{r_{i}}}\qquad(r_{i}\in\mathbb{N})\qquad\text{for~{}$1\leq i\leq m$}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⋯ italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_N ) for 1 ≤ italic_i ≤ italic_m

with the constraint λj(i)<isuperscriptsubscript𝜆𝑗𝑖𝑖\lambda_{j}^{(i)}<iitalic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT < italic_i for all i𝑖iitalic_i and j𝑗jitalic_j


Algorithm:

  • (Initialization)

    Introduce the exceptional type t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to denote the planted tree consisting of a single node (in other words, a leaf) and set m𝑚mitalic_m to 1111

  • (Main loop)

    For all planted trees of 𝒰𝒰\mathcal{U}caligraphic_U perform a depth-first traversal of the tree, starting from the planted root; during this recursive calculation, at each node n𝑛nitalic_n:

    1. (1)

      if the node is a leaf, return the type t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT

    2. (2)

      else, recursively determine the type associated with each child of n𝑛nitalic_n

    3. (3)

      If n𝑛nitalic_n is a not the planted root of the tree, write the subtree rooted at n𝑛nitalic_n as a (commutative) product π=xtλ1tλr𝜋𝑥subscript𝑡subscript𝜆1subscript𝑡subscript𝜆𝑟\pi=xt_{\lambda_{1}}\dotsm t_{\lambda_{r}}italic_π = italic_x italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT of the types obtained in the previous step

    4. (4)

      look up the uniquification table to check whether this product has already been assigned a type tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT

    5. (5)

      if not existent, increment m𝑚mitalic_m, create a new type tmsubscript𝑡𝑚t_{m}italic_t start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, remember its definition tm=πsubscript𝑡𝑚𝜋t_{m}=\piitalic_t start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_π, and assign tmsubscript𝑡𝑚t_{m}italic_t start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT to the product π𝜋\piitalic_π in the uniquification table.

    6. (6)

      return the type tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT if it was found by lookup, otherwise return tmsubscript𝑡𝑚t_{m}italic_t start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT

  • (Conclusion)

    Return m𝑚mitalic_m and the sequence of definitions of the form ti=πsubscript𝑡𝑖𝜋t_{i}=\piitalic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_π, for i=1,2,,m𝑖12𝑚i=1,2,\dots,mitalic_i = 1 , 2 , … , italic_m.

A.4. Disambiguating algorithm

  
The idea of the algorithm below is to consider each class of trees, tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, in turn, introducing its defining equation

ti=xtλ1(i)tλri(i)(r)subscript𝑡𝑖𝑥subscript𝑡superscriptsubscript𝜆1𝑖subscript𝑡superscriptsubscript𝜆subscript𝑟𝑖𝑖𝑟t_{i}=xt_{\lambda_{1}^{(i)}}\dotsm t_{\lambda_{r_{i}}^{(i)}}\qquad(r\in\mathbb{N})italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⋯ italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_r ∈ blackboard_N )

into the calculation, while maintaining (and refining) a partition

p=a0aL𝑝direct-sumsubscript𝑎0subscript𝑎𝐿p=a_{0}\oplus\dots\oplus a_{L}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ ⋯ ⊕ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT

of the total class of planted trees. To be able to do so, it is crucial that the recursive equation for tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT refers to classes tjsubscript𝑡𝑗t_{j}italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT with j<i𝑗𝑖j<iitalic_j < italic_i only, starting with the special class t0=psubscript𝑡0𝑝t_{0}=pitalic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_p, the full class of planted trees.

At any stage in the algorithm, the class of r𝑟ritalic_r-ary trees is given as the disjoint union of Cartesian products

λΛxtλ1tλrwhereΛ={λ:(λ)=r, 0λjL},subscriptdirect-sum𝜆Λ𝑥subscript𝑡subscript𝜆1subscript𝑡subscript𝜆𝑟whereΛconditional-set𝜆formulae-sequence𝜆𝑟 0subscript𝜆𝑗𝐿\bigoplus_{\lambda\in\Lambda}xt_{\lambda_{1}}\dotsm t_{\lambda_{r}}\qquad\text{where}\qquad\Lambda=\{\,\lambda:\ell(\lambda)=r,\ 0\leq\lambda_{j}\leq L\,\},⨁ start_POSTSUBSCRIPT italic_λ ∈ roman_Λ end_POSTSUBSCRIPT italic_x italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT where roman_Λ = { italic_λ : roman_ℓ ( italic_λ ) = italic_r , 0 ≤ italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≤ italic_L } ,

where (λ)𝜆\ell(\lambda)roman_ℓ ( italic_λ ) denotes the number of components in the tuple λ𝜆\lambdaitalic_λ. In the process of the algorithm below, each class tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT gets represented in a “polynomial” form like above, summed over a subset ΛΛ\Lambdaroman_Λ of the set of integer sequences λ=(λ1,,λr)𝜆subscript𝜆1subscript𝜆𝑟\lambda=(\lambda_{1},\dots,\lambda_{r})italic_λ = ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_λ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) of a given length r𝑟ritalic_r. Computing intersections and differences of classes means merely computing intersections and differences of the ΛΛ\Lambdaroman_Λ in their representations, because of the recursive structure of the input and of the algorithm itself.


Input:

  • A family 𝒮={t1,,tm}𝒮subscript𝑡1subscript𝑡𝑚\mathcal{S}=\{t_{1},\dots,t_{m}\}caligraphic_S = { italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } of classes of trees with recursive descriptions of the form

    ti=xtλ1(i)tλr(i)(r=(λ(i)))for 1imsubscript𝑡𝑖𝑥subscript𝑡superscriptsubscript𝜆1𝑖subscript𝑡superscriptsubscript𝜆𝑟𝑖𝑟superscript𝜆𝑖for 1imt_{i}=xt_{\lambda_{1}^{(i)}}\dotsm t_{\lambda_{r}^{(i)}}\qquad(r=\ell(\lambda^{(i)}))\qquad\text{for~{}$1\leq i\leq m$}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⋯ italic_t start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_r = roman_ℓ ( italic_λ start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT ) ) for 1 ≤ italic_i ≤ italic_m

    with the constraint λj(i)<isuperscriptsubscript𝜆𝑗𝑖𝑖\lambda_{j}^{(i)}<iitalic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT < italic_i for all i𝑖iitalic_i and j𝑗jitalic_j


Output:

  • an integer L𝐿Litalic_L implying a partition

    p=a0aL𝑝direct-sumsubscript𝑎0subscript𝑎𝐿p=a_{0}\oplus\dots\oplus a_{L}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ ⋯ ⊕ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT
  • a representation of each tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of the form

    ti=jIiajfor 0im and Ii{0,,L}subscript𝑡𝑖subscriptdirect-sum𝑗subscript𝐼𝑖subscript𝑎𝑗for 0im and Ii{0,,L}t_{i}=\bigoplus_{j\in I_{i}}a_{j}\qquad\text{for $0\leq i\leq m$ and~{}$I_{i}\subseteq\{0,\dots,L\}$}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT italic_j ∈ italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT for 0 ≤ italic_i ≤ italic_m and italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊆ { 0 , … , italic_L }
  • a recursive description of the aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of the form

    ai=λΛixaλ1aλ(λ)for 1iL,subscript𝑎𝑖subscriptdirect-sum𝜆subscriptΛ𝑖𝑥subscript𝑎subscript𝜆1subscript𝑎subscript𝜆𝜆for 1iLa_{i}=\bigoplus_{\lambda\in\Lambda_{i}}xa_{\lambda_{1}}\dotsm a_{\lambda_{\ell(\lambda)}}\qquad\text{for~{}$1\leq i\leq L$},italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT italic_λ ∈ roman_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_x italic_a start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT roman_ℓ ( italic_λ ) end_POSTSUBSCRIPT end_POSTSUBSCRIPT for 1 ≤ italic_i ≤ italic_L ,

    a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT being implicitly described as p(a1aL)𝑝direct-sumsubscript𝑎1subscript𝑎𝐿p\setminus(a_{1}\oplus\dots\oplus a_{L})italic_p ∖ ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ ⋯ ⊕ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )


Algorithm:

  • (Initialization)

    Start with the trivial partition p=a0𝑝subscript𝑎0p=a_{0}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT for L=0𝐿0L=0italic_L = 0, the single representation t0=a0subscript𝑡0subscript𝑎0t_{0}=a_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, that is, I0={0}subscript𝐼00I_{0}=\{0\}italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { 0 }.

  • (Main loop)

    For k𝑘kitalic_k from 1 to m𝑚mitalic_m do

    1. (1)

      replace each tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in the definition of tksubscript𝑡𝑘t_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT with its current representation in terms of the ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, expand, and set s𝑠sitalic_s to the result, so as to get a representation of tksubscript𝑡𝑘t_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT of the form

      s=λΛ(s)xaλ1aλ(λ)for some Λ(s)𝑠subscriptdirect-sum𝜆superscriptΛ𝑠𝑥subscript𝑎subscript𝜆1subscript𝑎subscript𝜆𝜆for some Λ(s)s=\bigoplus_{\lambda\in\Lambda^{(s)}}xa_{\lambda_{1}}\dotsm a_{\lambda_{\ell(\lambda)}}\qquad\text{for some~{}$\Lambda^{(s)}$}italic_s = ⨁ start_POSTSUBSCRIPT italic_λ ∈ roman_Λ start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_x italic_a start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT roman_ℓ ( italic_λ ) end_POSTSUBSCRIPT end_POSTSUBSCRIPT for some roman_Λ start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT
    2. (2)

      for i𝑖iitalic_i from 1 to L𝐿Litalic_L while s𝑠s\neq\emptysetitalic_s ≠ ∅ do

      1. (a)

        set b𝑏bitalic_b to aissubscript𝑎𝑖𝑠a_{i}\cap sitalic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_s by setting ΛsubscriptΛ\Lambda_{\cap}roman_Λ start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT to ΛiΛ(s)subscriptΛ𝑖superscriptΛ𝑠\Lambda_{i}\cap\Lambda^{(s)}roman_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ roman_Λ start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT

      2. (b)

        if b𝑏b\neq\emptysetitalic_b ≠ ∅, then do

        1. (i)

          set bsuperscript𝑏b^{\prime}italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT to aissubscript𝑎𝑖𝑠a_{i}\setminus sitalic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ italic_s

        2. (ii)

          if bsuperscript𝑏b^{\prime}\neq\emptysetitalic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ ∅, then

          1. (A)

            create a new ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT with description bsuperscript𝑏b^{\prime}italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT: increment n𝑛nitalic_n before setting aLsubscript𝑎𝐿a_{L}italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT to bsuperscript𝑏b^{\prime}italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, that is, before setting ΛLsubscriptΛ𝐿\Lambda_{L}roman_Λ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT to ΛiΛ(s)subscriptΛ𝑖superscriptΛ𝑠\Lambda_{i}\setminus\Lambda^{(s)}roman_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ roman_Λ start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT

          2. (B)

            split aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT into aiaLdirect-sumsubscript𝑎𝑖subscript𝑎𝐿a_{i}\oplus a_{L}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT in the representations of the tjsubscript𝑡𝑗t_{j}italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, that is, add n𝑛nitalic_n into each set Ijsubscript𝐼𝑗I_{j}italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT containing i𝑖iitalic_i

          3. (C)

            split aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT into aiaLdirect-sumsubscript𝑎𝑖subscript𝑎𝐿a_{i}\oplus a_{L}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT in the descriptions of the ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, b𝑏bitalic_b, and s𝑠sitalic_s, that is, for each sequence in each of the ΛjsubscriptΛ𝑗\Lambda_{j}roman_Λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, ΛsubscriptΛ\Lambda_{\cap}roman_Λ start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT, and Λ(s)superscriptΛ𝑠\Lambda^{(s)}roman_Λ start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT, add sequences with i𝑖iitalic_i replaced by L𝐿Litalic_L when the sequence involves i𝑖iitalic_i (if i𝑖iitalic_i occurs more than once, then replace i𝑖iitalic_i by i𝑖iitalic_i or L𝐿Litalic_L in all possible ways)

          4. (D)

            set aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to b𝑏bitalic_b by setting ΛisubscriptΛ𝑖\Lambda_{i}roman_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to ΛsubscriptΛ\Lambda_{\cap}roman_Λ start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT

        3. (iii)

          set s𝑠sitalic_s to sb𝑠𝑏s\setminus bitalic_s ∖ italic_b, which is also sai𝑠subscript𝑎𝑖s\setminus a_{i}italic_s ∖ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, and update Λ(s)superscriptΛ𝑠\Lambda^{(s)}roman_Λ start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT by setting it to Λ(s)ΛisuperscriptΛ𝑠subscriptΛ𝑖\Lambda^{(s)}\setminus\Lambda_{i}roman_Λ start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT ∖ roman_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT

    3. (3)

      if s𝑠s\neq\emptysetitalic_s ≠ ∅, then

      1. (a)

        create a new ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT with description s𝑠sitalic_s: increment L𝐿Litalic_L before setting aLsubscript𝑎𝐿a_{L}italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT to s𝑠sitalic_s, that is, before setting ΛLsubscriptΛ𝐿\Lambda_{L}roman_Λ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT to Λ(s)superscriptΛ𝑠\Lambda^{(s)}roman_Λ start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT

      2. (b)

        split a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT into a0aLdirect-sumsubscript𝑎0subscript𝑎𝐿a_{0}\oplus a_{L}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT in the representations of the tjsubscript𝑡𝑗t_{j}italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, that is, add L𝐿Litalic_L into each set Ijsubscript𝐼𝑗I_{j}italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT containing 00

      3. (c)

        split a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT into a0aLdirect-sumsubscript𝑎0subscript𝑎𝐿a_{0}\oplus a_{L}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT in the descriptions of the ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, that is, for each sequence in each of the ΛjsubscriptΛ𝑗\Lambda_{j}roman_Λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, add sequences with 00 replaced by n𝑛nitalic_n when the sequence involves 00 (if 00 occurs more than once, then replace 00 by 00 or L𝐿Litalic_L in all possible ways)

    4. (4)

      represent tksubscript𝑡𝑘t_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT as the union of all those aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPTs that have contributed a non-empty b𝑏bitalic_b at step (2b) and of aLsubscript𝑎𝐿a_{L}italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT if a new ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT was created at step (3a), that is, create the corresponding set Iksubscript𝐼𝑘I_{k}italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT consisting of the contributing i𝑖iitalic_is, together with L𝐿Litalic_L if relevant

  • (Final step)

    Return L𝐿Litalic_L, the representations of the tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for 1im1𝑖𝑚1\leq i\leq m1 ≤ italic_i ≤ italic_m, the recursive descriptions of the aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for 1iL1𝑖𝐿1\leq i\leq L1 ≤ italic_i ≤ italic_L

We will explicitly show the stages through which the algorithm goes when running with the input (18)italic-(18italic-)\eqref{eq:dag}italic_( italic_). For readability, we will keep expressions in factored form.

  • k=1𝑘1k=1italic_k = 1:

    from t1=xa03subscript𝑡1𝑥superscriptsubscript𝑎03t_{1}=xa_{0}^{3}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, we derive t1=a1subscript𝑡1subscript𝑎1t_{1}=a_{1}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and a1=x(a0a1)3subscript𝑎1𝑥superscriptdirect-sumsubscript𝑎0subscript𝑎13a_{1}=x(a_{0}\oplus a_{1})^{3}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT.

  • k=2𝑘2k=2italic_k = 2:

    from t2=x(a0a1)a1subscript𝑡2𝑥direct-sumsubscript𝑎0subscript𝑎1subscript𝑎1t_{2}=x(a_{0}\oplus a_{1})a_{1}italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, we derive t1=a1,t2=a2formulae-sequencesubscript𝑡1subscript𝑎1subscript𝑡2subscript𝑎2t_{1}=a_{1},\ t_{2}=a_{2}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and a1=xp3,a2=xpa1formulae-sequencesubscript𝑎1𝑥superscript𝑝3subscript𝑎2𝑥𝑝subscript𝑎1a_{1}=xp^{3},\ a_{2}=xpa_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x italic_p italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, where p=a0a1a2𝑝direct-sumsubscript𝑎0subscript𝑎1subscript𝑎2p=a_{0}\oplus a_{1}\oplus a_{2}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

  • k=3𝑘3k=3italic_k = 3:

    from t3=xa12subscript𝑡3𝑥superscriptsubscript𝑎12t_{3}=xa_{1}^{2}italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, we derive t1=a1,t2=a2a3,t3=a2formulae-sequencesubscript𝑡1subscript𝑎1formulae-sequencesubscript𝑡2direct-sumsubscript𝑎2subscript𝑎3subscript𝑡3subscript𝑎2t_{1}=a_{1},\ t_{2}=a_{2}\oplus a_{3},\ t_{3}=a_{2}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and a1=xp3,a2=xa12,a3=x(a0a2a3)a1formulae-sequencesubscript𝑎1𝑥superscript𝑝3formulae-sequencesubscript𝑎2𝑥superscriptsubscript𝑎12subscript𝑎3𝑥direct-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎1a_{1}=xp^{3},\ a_{2}=xa_{1}^{2},\ a_{3}=x(a_{0}\oplus a_{2}\oplus a_{3})a_{1}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, where p=a0a1a2a3𝑝direct-sumsubscript𝑎0subscript𝑎1subscript𝑎2subscript𝑎3p=a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{3}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.

  • k=4𝑘4k=4italic_k = 4:

    from t4=x(a0a1a2a3)4subscript𝑡4𝑥superscriptdirect-sumsubscript𝑎0subscript𝑎1subscript𝑎2subscript𝑎34t_{4}=x(a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{3})^{4}italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, we derive t1=a1,t2=a2a3,t3=a2,t4=a4formulae-sequencesubscript𝑡1subscript𝑎1formulae-sequencesubscript𝑡2direct-sumsubscript𝑎2subscript𝑎3formulae-sequencesubscript𝑡3subscript𝑎2subscript𝑡4subscript𝑎4t_{1}=a_{1},\ t_{2}=a_{2}\oplus a_{3},\ t_{3}=a_{2},\ t_{4}=a_{4}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT and a1=xp3,a2=xa12,a3=x(a0a2a3a4)a1,a4=xp4formulae-sequencesubscript𝑎1𝑥superscript𝑝3formulae-sequencesubscript𝑎2𝑥superscriptsubscript𝑎12formulae-sequencesubscript𝑎3𝑥direct-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎4subscript𝑎1subscript𝑎4𝑥superscript𝑝4a_{1}=xp^{3},\ a_{2}=xa_{1}^{2},\ a_{3}=x(a_{0}\oplus a_{2}\oplus a_{3}\oplus a_{4})a_{1},\ a_{4}=xp^{4}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_x italic_p start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, where p=a0a1a2a3a4𝑝direct-sumsubscript𝑎0subscript𝑎1subscript𝑎2subscript𝑎3subscript𝑎4p=a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{3}\oplus a_{4}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT.

  • k=5𝑘5k=5italic_k = 5:

    from t5=x(a0a1a2a3a4)a4subscript𝑡5𝑥direct-sumsubscript𝑎0subscript𝑎1subscript𝑎2subscript𝑎3subscript𝑎4subscript𝑎4t_{5}=x(a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{3}\oplus a_{4})a_{4}italic_t start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, we derive t1=a1,t2=a2a3a5,t3=a2,t4=a4,t5=a3a6formulae-sequencesubscript𝑡1subscript𝑎1formulae-sequencesubscript𝑡2direct-sumsubscript𝑎2subscript𝑎3subscript𝑎5formulae-sequencesubscript𝑡3subscript𝑎2formulae-sequencesubscript𝑡4subscript𝑎4subscript𝑡5direct-sumsubscript𝑎3subscript𝑎6t_{1}=a_{1},\ t_{2}=a_{2}\oplus a_{3}\oplus a_{5},\ t_{3}=a_{2},\ t_{4}=a_{4},\ t_{5}=a_{3}\oplus a_{6}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT and a1=xp3,a2=xa12,a3=xa1a4,a4=xp4,a5=x(a0a2a3a5a6)a1,a6=x(a0a2a3a4a5a6)a4formulae-sequencesubscript𝑎1𝑥superscript𝑝3formulae-sequencesubscript𝑎2𝑥superscriptsubscript𝑎12formulae-sequencesubscript𝑎3𝑥subscript𝑎1subscript𝑎4formulae-sequencesubscript𝑎4𝑥superscript𝑝4formulae-sequencesubscript𝑎5𝑥direct-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎1subscript𝑎6𝑥direct-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎4subscript𝑎5subscript𝑎6subscript𝑎4a_{1}=xp^{3},\ a_{2}=xa_{1}^{2},\ a_{3}=xa_{1}a_{4},\ a_{4}=xp^{4},\ a_{5}=x(a_{0}\oplus a_{2}\oplus a_{3}\oplus a_{5}\oplus a_{6})a_{1},\ a_{6}=x(a_{0}\oplus a_{2}\oplus a_{3}\oplus a_{4}\oplus a_{5}\oplus a_{6})a_{4}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_x italic_p start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, where p=a0a1a2a3a4a5a6𝑝direct-sumsubscript𝑎0subscript𝑎1subscript𝑎2subscript𝑎3subscript𝑎4subscript𝑎5subscript𝑎6p=a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{3}\oplus a_{4}\oplus a_{5}\oplus a_{6}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT.

  • k=6𝑘6k=6italic_k = 6:

    from t6=xa42subscript𝑡6𝑥superscriptsubscript𝑎42t_{6}=xa_{4}^{2}italic_t start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, we derive t1=a1,t2=a2a3a5,t3=a2,t4=a4,t5=a3a6a7,t6=a6formulae-sequencesubscript𝑡1subscript𝑎1formulae-sequencesubscript𝑡2direct-sumsubscript𝑎2subscript𝑎3subscript𝑎5formulae-sequencesubscript𝑡3subscript𝑎2formulae-sequencesubscript𝑡4subscript𝑎4formulae-sequencesubscript𝑡5direct-sumsubscript𝑎3subscript𝑎6subscript𝑎7subscript𝑡6subscript𝑎6t_{1}=a_{1},\ t_{2}=a_{2}\oplus a_{3}\oplus a_{5},\ t_{3}=a_{2},\ t_{4}=a_{4},\ t_{5}=a_{3}\oplus a_{6}\oplus a_{7},\ t_{6}=a_{6}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT and a1=xp3,a2=xa12,a3=xa1a4,a4=xp4,a5=x(a0a2a3a5a6a7)a1,a6=xa42,a7=x(a0a2a3a5a6a7)a4formulae-sequencesubscript𝑎1𝑥superscript𝑝3formulae-sequencesubscript𝑎2𝑥superscriptsubscript𝑎12formulae-sequencesubscript𝑎3𝑥subscript𝑎1subscript𝑎4formulae-sequencesubscript𝑎4𝑥superscript𝑝4formulae-sequencesubscript𝑎5𝑥direct-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎1formulae-sequencesubscript𝑎6𝑥superscriptsubscript𝑎42subscript𝑎7𝑥direct-sumsubscript𝑎0subscript𝑎2subscript𝑎3subscript𝑎5subscript𝑎6subscript𝑎7subscript𝑎4a_{1}=xp^{3},\ a_{2}=xa_{1}^{2},\ a_{3}=xa_{1}a_{4},\ a_{4}=xp^{4},\ a_{5}=x(a_{0}\oplus a_{2}\oplus a_{3}\oplus a_{5}\oplus a_{6}\oplus a_{7})a_{1},\ a_{6}=xa_{4}^{2},\ a_{7}=x(a_{0}\oplus a_{2}\oplus a_{3}\oplus a_{5}\oplus a_{6}\oplus a_{7})a_{4}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_x italic_p start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT = italic_x italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT = italic_x ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, where p=a0a1a2a3a4a5a6a7𝑝direct-sumsubscript𝑎0subscript𝑎1subscript𝑎2subscript𝑎3subscript𝑎4subscript𝑎5subscript𝑎6subscript𝑎7p=a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{3}\oplus a_{4}\oplus a_{5}\oplus a_{6}\oplus a_{7}italic_p = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ⊕ italic_a start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT.

A.5. Calculation of K(l0,,lL)𝐾subscript𝑙0subscript𝑙𝐿K(l_{0},\dots,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ): CountRootOccurrences


Input: non-planar planted trees τ𝜏\tauitalic_τ, τ1subscript𝜏1\tau_{1}italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, …, τksubscript𝜏𝑘\tau_{k}italic_τ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT

Output: the number of occurrences of any of the τisubscript𝜏𝑖\tau_{i}italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT at the root of τ𝜏\tauitalic_τ

Algorithm:

  1. (1)

    fix one element πsuperscript𝜋\pi^{\prime}italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT from GeneralToPlanar(τ)GeneralToPlanar𝜏\mathrm{GeneralToPlanar}(\tau)roman_GeneralToPlanar ( italic_τ ) (see algorithm A.1)

  2. (2)

    for each i𝑖iitalic_i between 1 and k𝑘kitalic_k, compute Pi=GeneralToPlanar(τ)subscript𝑃𝑖GeneralToPlanar𝜏P_{i}=\mathrm{GeneralToPlanar}(\tau)italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_GeneralToPlanar ( italic_τ )

  3. (3)

    count and return the number of pairs (πi,π)subscript𝜋𝑖superscript𝜋(\pi_{i},\pi^{\prime})( italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) such that πisubscript𝜋𝑖\pi_{i}italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is element of Pisubscript𝑃𝑖P_{i}italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and πisubscript𝜋𝑖\pi_{i}italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT occurs at the root of πsuperscript𝜋\pi^{\prime}italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT

As an example we calculate K(0,1,0,1,0,0,1,0)𝐾01010010K(0,1,0,1,0,0,1,0)italic_K ( 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 ). This corresponds to calculating the number of additional occurrences in the class xa1a3a6𝑥subscript𝑎1subscript𝑎3subscript𝑎6xa_{1}a_{3}a_{6}italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT. The input trees τ,τ1,τ2,τ3𝜏subscript𝜏1subscript𝜏2subscript𝜏3\tau,\tau_{1},\tau_{2},\tau_{3}italic_τ , italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT are shown in Figure 14. Here τ𝜏\tauitalic_τ corresponds to the class xa1a3a6𝑥subscript𝑎1subscript𝑎3subscript𝑎6xa_{1}a_{3}a_{6}italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT and τ1,τ2,τ3subscript𝜏1subscript𝜏2subscript𝜏3\tau_{1},\tau_{2},\tau_{3}italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT correspond to the three possible ways of planting the example pattern.

Refer to caption
Figure 14. Input trees τ,τ1,τ2,τ3𝜏subscript𝜏1subscript𝜏2subscript𝜏3\tau,\tau_{1},\tau_{2},\tau_{3}italic_τ , italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT

We take as fixed planar embedding πsuperscript𝜋\pi^{\prime}italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT of τ𝜏\tauitalic_τ the embedding of Figure 14. We now iterate over the different planar embeddings π1subscript𝜋1\pi_{1}italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT of τ1subscript𝜏1\tau_{1}italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (6 of them), π2subscript𝜋2\pi_{2}italic_π start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of τ2subscript𝜏2\tau_{2}italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (2 of them), and π3subscript𝜋3\pi_{3}italic_π start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT of τ3subscript𝜏3\tau_{3}italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT (8 of them), and determine for each πisubscript𝜋𝑖\pi_{i}italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (i{1,2,3}𝑖123i\in\{1,2,3\}italic_i ∈ { 1 , 2 , 3 }) whether it occurs at the root of πsuperscript𝜋\pi^{\prime}italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Consider for example the four embeddings shown in Figure 10 (three embeddings of τ1subscript𝜏1\tau_{1}italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, one embedding of τ3subscript𝜏3\tau_{3}italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT). The leftmost embedding matches πsuperscript𝜋\pi^{\prime}italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, the one next to it as well. The third one does not match πsuperscript𝜋\pi^{\prime}italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, because the node with out-degree four is in the wrong position. The rightmost embedding clearly does not match either. By considering all embeddings and counting the matches we get k=K(0,1,0,1,0,0,1,0)=3𝑘𝐾010100103k=K(0,1,0,1,0,0,1,0)=3italic_k = italic_K ( 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 ) = 3.

The algorithm calculates the correct value of k𝑘kitalic_k, because the partition consisting of the classes aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is sufficiently fine. From this follows that every match above of a planar embedding really gives rise to exactly one additional pattern occurrence. See the considerations made at the beginning of this appendix.

By now the transformation to a systems of equations is easy. We get the terms by replacing a term xaj1ajs𝑥subscript𝑎subscript𝑗1subscript𝑎subscript𝑗𝑠xa_{j_{1}}\dotsm a_{j_{s}}italic_x italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT in the recursive description of ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT by a term xyj1yjsuK(l0,,lL)/l0!lL!𝑥subscript𝑦subscript𝑗1subscript𝑦subscript𝑗𝑠superscript𝑢𝐾subscript𝑙0subscript𝑙𝐿subscript𝑙0subscript𝑙𝐿xy_{j_{1}}\dotsm y_{j_{s}}u^{K(l_{0},\dots,l_{L})}/l_{0}!\dots l_{L}!italic_x italic_y start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_y start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT / italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ! … italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT !. Here it is assumed that terms that represent the same tree classes (like xa1a2𝑥subscript𝑎1subscript𝑎2xa_{1}a_{2}italic_x italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and xa2a1𝑥subscript𝑎2subscript𝑎1xa_{2}a_{1}italic_x italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT) are identified before. It is clear that there are only finitely many terms for which K(l0,,lL)𝐾subscript𝑙0subscript𝑙𝐿K(l_{0},\dots,l_{L})italic_K ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_l start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) might be non-zero a priori.

Appendix B Asymptotics of Analytic Systems

The following theorem is a slightly modified version of the main theorem from [Drm97]. We denote the transpose of a vector v𝑣vitalic_v by vTsuperscript𝑣Tv^{\mathrm{T}}italic_v start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT. Let 𝐅(x,𝐲,𝐮)=(a10(x,𝐲,𝐮),,FN(x,𝐲,𝐮))T𝐅𝑥𝐲𝐮superscriptsubscript𝑎10𝑥𝐲𝐮subscript𝐹𝑁𝑥𝐲𝐮T{\bf F}(x,{\bf y},{\bf u})=(a_{10}(x,{\bf y},{\bf u}),\ldots,F_{N}(x,{\bf y},{\bf u}))^{\mathrm{T}}bold_F ( italic_x , bold_y , bold_u ) = ( italic_a start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) , … , italic_F start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT be a column vector of functions Fj(x,𝐲,𝐮)subscript𝐹𝑗𝑥𝐲𝐮F_{j}(x,{\bf y},{\bf u})italic_F start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ), 1jN1𝑗𝑁1\leq j\leq N1 ≤ italic_j ≤ italic_N, with complex variables x𝑥xitalic_x, 𝐲=(y1,,yN)T𝐲superscriptsubscript𝑦1subscript𝑦𝑁T{\bf y}=(y_{1},\ldots,y_{N})^{\mathrm{T}}bold_y = ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT, 𝐮=(u1,,uk)T𝐮superscriptsubscript𝑢1subscript𝑢𝑘T{\bf u}=(u_{1},\ldots,u_{k})^{\mathrm{T}}bold_u = ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT which are analytic around 00 and satisfy Fj(0,𝟎,𝟎)=0subscript𝐹𝑗0000F_{j}(0,{\bf 0},{\bf 0})=0italic_F start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 0 , bold_0 , bold_0 ) = 0 for 1jN1𝑗𝑁1\leq j\leq N1 ≤ italic_j ≤ italic_N. We are interested in the analytic solution 𝐲=𝐲(x,𝐮)=(y1(x,𝐮),,yN(x,𝐮))T𝐲𝐲𝑥𝐮superscriptsubscript𝑦1𝑥𝐮subscript𝑦𝑁𝑥𝐮T{\bf y}={\bf y}(x,{\bf u})=(y_{1}(x,{\bf u}),\ldots,y_{N}(x,{\bf u}))^{\mathrm{T}}bold_y = bold_y ( italic_x , bold_u ) = ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , bold_u ) , … , italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x , bold_u ) ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT of the functional equation

(20) 𝐲=𝐅(x,𝐲,𝐮)𝐲𝐅𝑥𝐲𝐮{\bf y}={\bf F}(x,{\bf y},{\bf u})bold_y = bold_F ( italic_x , bold_y , bold_u )

with 𝐲(0,𝟎)=𝟎𝐲000{\bf y}(0,{\bf 0})={\bf 0}bold_y ( 0 , bold_0 ) = bold_0, i.e., we demand that the (unknown) functions yj=yj(x,𝐮)subscript𝑦𝑗subscript𝑦𝑗𝑥𝐮y_{j}=y_{j}(x,{\bf u})italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_u ), 1jN1𝑗𝑁1\leq j\leq N1 ≤ italic_j ≤ italic_N, satisfy the system of functional equations

y1subscript𝑦1\displaystyle y_{1}italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =F1(x,y1,y2,,yN,𝐮),absentsubscript𝐹1𝑥subscript𝑦1subscript𝑦2subscript𝑦𝑁𝐮\displaystyle=F_{1}(x,y_{1},y_{2},\ldots,y_{N},{\bf u}),= italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , bold_u ) ,
y2subscript𝑦2\displaystyle y_{2}italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =F2(x,y1,y2,,yN,𝐮),absentsubscript𝐹2𝑥subscript𝑦1subscript𝑦2subscript𝑦𝑁𝐮\displaystyle=F_{2}(x,y_{1},y_{2},\ldots,y_{N},{\bf u}),= italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , bold_u ) ,
\displaystyle\,\vdots
yNsubscript𝑦𝑁\displaystyle y_{N}italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT =FN(x,y1,y2,,yN,𝐮).absentsubscript𝐹𝑁𝑥subscript𝑦1subscript𝑦2subscript𝑦𝑁𝐮\displaystyle=F_{N}(x,y_{1},y_{2},\ldots,y_{N},{\bf u}).= italic_F start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , bold_u ) .

It is convenient to define the notion of a dependency (di)graph G𝐅=(V,E)subscript𝐺𝐅𝑉𝐸G_{{\bf F}}=(V,E)italic_G start_POSTSUBSCRIPT bold_F end_POSTSUBSCRIPT = ( italic_V , italic_E ) for such a system of functional equations 𝐲=𝐅(x,𝐲,𝐮)𝐲𝐅𝑥𝐲𝐮{\bf y}={\bf F}(x,{\bf y},{\bf u})bold_y = bold_F ( italic_x , bold_y , bold_u ). The vertices V={y1,y2,,yN}𝑉subscript𝑦1subscript𝑦2subscript𝑦𝑁V=\{y_{1},y_{2},\ldots,y_{N}\}italic_V = { italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT } are just the unknown functions and an ordered pair (yi,yj)subscript𝑦𝑖subscript𝑦𝑗(y_{i},y_{j})( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) is contained in the edge set E𝐸Eitalic_E if and only if Fi(x,𝐲,𝐮)subscript𝐹𝑖𝑥𝐲𝐮F_{i}(x,{\bf y},{\bf u})italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) really depends on yjsubscript𝑦𝑗y_{j}italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT.

If the functions Fj(x,𝐲,𝐮)subscript𝐹𝑗𝑥𝐲𝐮F_{j}(x,{\bf y},{\bf u})italic_F start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) have non-negative Taylor coefficients then it is easy to see that the solutions yj(x,𝐮)subscript𝑦𝑗𝑥𝐮y_{j}(x,{\bf u})italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_u ) have the same property. (One only has to solve the system iteratively by setting 𝐲0(x,𝐮)=0subscript𝐲0𝑥𝐮0{\bf y}_{0}(x,{\bf u})=0bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , bold_u ) = 0 and 𝐲i+1(x,𝐮)=𝐅(x,𝐲i(x,𝐮),𝐮)subscript𝐲𝑖1𝑥𝐮𝐅𝑥subscript𝐲𝑖𝑥𝐮𝐮{\bf y}_{i+1}(x,{\bf u})={\bf F}(x,{\bf y}_{i}(x,{\bf u}),{\bf u})bold_y start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ( italic_x , bold_u ) = bold_F ( italic_x , bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , bold_u ) , bold_u ) for i0𝑖0i\geq 0italic_i ≥ 0. The limit 𝐲(x,𝐮)=limi𝐲i(x,𝐮)𝐲𝑥𝐮subscript𝑖subscript𝐲𝑖𝑥𝐮{\bf y}(x,{\bf u})=\lim_{i\to\infty}{\bf y}_{i}(x,{\bf u})bold_y ( italic_x , bold_u ) = roman_lim start_POSTSUBSCRIPT italic_i → ∞ end_POSTSUBSCRIPT bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , bold_u ) is the (unique) solution of the system above.)

Now suppose that G(x,𝐲,𝐮)𝐺𝑥𝐲𝐮G(x,{\bf y},{\bf u})italic_G ( italic_x , bold_y , bold_u ) is another analytic function with non-negative Taylor coefficients. Then G(x,𝐲(x,𝐮),𝐮)𝐺𝑥𝐲𝑥𝐮𝐮G(x,{\bf y}(x,{\bf u}),{\bf u})italic_G ( italic_x , bold_y ( italic_x , bold_u ) , bold_u ) has a power series expansion

G(x,𝐲(x,𝐮),𝐮)=n,𝐦cn,𝐦xn𝐮𝐦𝐺𝑥𝐲𝑥𝐮𝐮subscript𝑛𝐦subscript𝑐𝑛𝐦superscript𝑥𝑛superscript𝐮𝐦G(x,{\bf y}(x,{\bf u}),{\bf u})=\sum_{n,{\bf m}}c_{n,{\bf m}}x^{n}{\bf u}^{{\bf m}}italic_G ( italic_x , bold_y ( italic_x , bold_u ) , bold_u ) = ∑ start_POSTSUBSCRIPT italic_n , bold_m end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n , bold_m end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT bold_u start_POSTSUPERSCRIPT bold_m end_POSTSUPERSCRIPT

with non-negative coefficients cn,𝐦subscript𝑐𝑛𝐦c_{n,{\bf m}}italic_c start_POSTSUBSCRIPT italic_n , bold_m end_POSTSUBSCRIPT. In fact, we assume that for every nn0𝑛subscript𝑛0n\geq n_{0}italic_n ≥ italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT there exists 𝐦𝐦{\bf m}bold_m such that cn,𝐦>0subscript𝑐𝑛𝐦0c_{n,{\bf m}}>0italic_c start_POSTSUBSCRIPT italic_n , bold_m end_POSTSUBSCRIPT > 0.

Let 𝐗nsubscript𝐗𝑛{\bf X}_{n}bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT (nn0)𝑛subscript𝑛0(n\geq n_{0})( italic_n ≥ italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) denote an N𝑁Nitalic_N-dimensional discrete random vector with

(21) 𝐏𝐫[𝐗n=𝐦]:=cn,𝐦cn,assign𝐏𝐫delimited-[]subscript𝐗𝑛𝐦subscript𝑐𝑛𝐦subscript𝑐𝑛{\bf Pr}[{\bf X}_{n}={\bf m}]:=\frac{c_{n,{\bf m}}}{c_{n}},bold_Pr [ bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_m ] := divide start_ARG italic_c start_POSTSUBSCRIPT italic_n , bold_m end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG ,

where

cn=𝐦cn,𝐦subscript𝑐𝑛subscript𝐦subscript𝑐𝑛𝐦c_{n}=\sum_{{\bf m}}c_{n,{\bf m}}italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT bold_m end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n , bold_m end_POSTSUBSCRIPT

are the coefficients of

G(x,𝐲(x,𝟏),𝟏)=n0cnxn.𝐺𝑥𝐲𝑥11subscript𝑛0subscript𝑐𝑛superscript𝑥𝑛G(x,{\bf y}(x,{\bf 1}),{\bf 1})=\sum_{n\geq 0}c_{n}x^{n}.italic_G ( italic_x , bold_y ( italic_x , bold_1 ) , bold_1 ) = ∑ start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .

The following theorem shows that (under suitable analyticity conditions) 𝐗nsubscript𝐗𝑛{\bf X}_{n}bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT has a Gaussian limiting distribution.

Theorem 2.

Let 𝐅(x,𝐲,𝐮)=(a1(x,𝐲,𝐮),,FN(x,𝐲,𝐮))T𝐅𝑥𝐲𝐮superscriptsubscript𝑎1𝑥𝐲𝐮normal-…subscript𝐹𝑁𝑥𝐲𝐮normal-T{\bf F}(x,{\bf y},{\bf u})=(a_{1}(x,{\bf y},{\bf u}),\ldots,F_{N}(x,{\bf y},{\bf u}))^{\mathrm{T}}bold_F ( italic_x , bold_y , bold_u ) = ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) , … , italic_F start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT be functions analytic around x=0𝑥0x=0italic_x = 0,  𝐲=(y1,,yN)T=𝟎𝐲superscriptsubscript𝑦1normal-…subscript𝑦𝑁normal-T0{\bf y}=(y_{1},\ldots,y_{N})^{\mathrm{T}}={\bf 0}bold_y = ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT = bold_0,  𝐮=(u1,,uk)T=𝟎𝐮superscriptsubscript𝑢1normal-…subscript𝑢𝑘normal-T0{\bf u}=(u_{1},\ldots,u_{k})^{\mathrm{T}}={\bf 0}bold_u = ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT = bold_0, whose Taylor coefficients are all non-negative, such that 𝐅(0,𝐲,𝐮)=𝟎𝐅0𝐲𝐮0{\bf F}(0,{\bf y},{\bf u})={\bf 0}bold_F ( 0 , bold_y , bold_u ) = bold_0,  𝐅(x,𝟎,𝐮)𝟎𝐅𝑥0𝐮0{\bf F}(x,{\bf 0},{\bf u})\neq{\bf 0}bold_F ( italic_x , bold_0 , bold_u ) ≠ bold_0,  𝐅x(x,𝐲,𝐮)𝟎subscript𝐅𝑥𝑥𝐲𝐮0{\bf F}_{x}(x,{\bf y},{\bf u})\not={\bf 0}bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) ≠ bold_0, and such that there exists j𝑗jitalic_j with 𝐅yjyj(x,𝐲,𝐮)𝟎subscript𝐅subscript𝑦𝑗subscript𝑦𝑗𝑥𝐲𝐮0{\bf F}_{y_{j}y_{j}}(x,{\bf y},{\bf u})\neq{\bf 0}bold_F start_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) ≠ bold_0. Furthermore assume that the region of convergence of 𝐅𝐅{\bf F}bold_F is large enough that there exists a non-negative solution x=x0𝑥subscript𝑥0x=x_{0}italic_x = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT,  𝐲=𝐲0𝐲subscript𝐲0{\bf y}={\bf y}_{0}bold_y = bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT of the system of equations

𝐲𝐲\displaystyle{\bf y}bold_y =𝐅(x,𝐲,𝟏),absent𝐅𝑥𝐲1\displaystyle={\bf F}(x,{\bf y},{\bf 1}),= bold_F ( italic_x , bold_y , bold_1 ) ,
00\displaystyle 0 =det(𝐈𝐅𝐲(x,𝐲,𝟏)),absent𝐈subscript𝐅𝐲𝑥𝐲1\displaystyle=\det({\bf I}-{\bf F}_{\bf y}(x,{\bf y},{\bf 1})),= roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x , bold_y , bold_1 ) ) ,

inside it. Let

𝐲=𝐲(x,𝐮)=(y1(x,𝐮),,yN(x,𝐮))T𝐲𝐲𝑥𝐮superscriptsubscript𝑦1𝑥𝐮subscript𝑦𝑁𝑥𝐮T{\bf y}={\bf y}(x,{\bf u})=(y_{1}(x,{\bf u}),\ldots,y_{N}(x,{\bf u}))^{\mathrm{T}}bold_y = bold_y ( italic_x , bold_u ) = ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , bold_u ) , … , italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x , bold_u ) ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT

denote the analytic solutions of the system

(22) 𝐲=𝐅(x,𝐲,𝐮)𝐲𝐅𝑥𝐲𝐮{\bf y}={\bf F}(x,{\bf y},{\bf u})bold_y = bold_F ( italic_x , bold_y , bold_u )

with 𝐲(0,𝐮)=𝟎𝐲0𝐮0{\bf y}(0,{\bf u})=\bf 0bold_y ( 0 , bold_u ) = bold_0 and assume that dn,j>0subscript𝑑𝑛𝑗0d_{n,j}>0italic_d start_POSTSUBSCRIPT italic_n , italic_j end_POSTSUBSCRIPT > 0 (1jN)1𝑗𝑁(1\leq j\leq N)( 1 ≤ italic_j ≤ italic_N ) for nn1𝑛subscript𝑛1n\geq n_{1}italic_n ≥ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, where yj(x,𝟏)=n0dn,jxnsubscript𝑦𝑗𝑥1subscript𝑛0subscript𝑑𝑛𝑗superscript𝑥𝑛y_{j}(x,{\bf 1})=\sum_{n\geq 0}d_{n,j}x^{n}italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_1 ) = ∑ start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_n , italic_j end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Moreover, let G(x,𝐲,𝐮)𝐺𝑥𝐲𝐮G(x,{\bf y},{\bf u})italic_G ( italic_x , bold_y , bold_u ) denote an analytic function with non-negative Taylor coefficients such that the point (x0,𝐲(x0,𝟏),𝟏)subscript𝑥0𝐲subscript𝑥011(x_{0},{\bf y}(x_{0},{\bf 1}),{\bf 1})( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) , bold_1 ) is contained in the region of convergence. Finally, let random vectors 𝐗nsubscript𝐗𝑛{\bf X}_{n}bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT (nn0)𝑛subscript𝑛0(n\geq n_{0})( italic_n ≥ italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) be defined by (21).

If the dependency graph G𝐅=(V,E)subscript𝐺𝐅𝑉𝐸G_{\bf F}=(V,E)italic_G start_POSTSUBSCRIPT bold_F end_POSTSUBSCRIPT = ( italic_V , italic_E ) of the system (22) in the unknown functions y1(x,𝐮),,subscript𝑦1𝑥𝐮normal-…y_{1}(x,{\bf u}),\ldots,italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , bold_u ) , … , yN(x,𝐮)subscript𝑦𝑁𝑥𝐮y_{N}(x,{\bf u})italic_y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x , bold_u ) is strongly connected then the sequence of random vectors 𝐗nsubscript𝐗𝑛{\bf X}_{n}bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT admits a Gaussian limiting distribution with mean value

𝐄𝐗n=𝝁n+O(1)(n)𝐄subscript𝐗𝑛𝝁𝑛𝑂1𝑛{\bf E}\,{\bf X}_{n}=\mbox{\boldmath{$\mu$}}\,n+O(1)\qquad(n\to\infty)bold_E bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_italic_μ italic_n + italic_O ( 1 ) ( italic_n → ∞ )

and covariance matrix

𝐂𝐨𝐯(𝐗n,𝐗n)=𝚺n+O(1)(n).𝐂𝐨𝐯subscript𝐗𝑛subscript𝐗𝑛𝚺𝑛𝑂1𝑛{\bf Cov}({\bf X}_{n},{\bf X}_{n})={\bf\Sigma}\,n+O(1)\qquad(n\to\infty).bold_Cov ( bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = bold_Σ italic_n + italic_O ( 1 ) ( italic_n → ∞ ) .

The row vector 𝛍𝛍\mubold_italic_μ is given by

𝝁=x𝐮(𝟏)x(𝟏),𝝁subscript𝑥𝐮1𝑥1\mbox{\boldmath{$\mu$}}=-\frac{x_{\bf u}({\bf 1})}{x({\bf 1})},bold_italic_μ = - divide start_ARG italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( bold_1 ) end_ARG start_ARG italic_x ( bold_1 ) end_ARG ,

and the matrix 𝚺𝚺\bf\Sigmabold_Σ by

(23) 𝚺=𝐱𝐮𝐮(𝟏)𝐱(𝟏)+𝝁T𝝁+diag(𝝁),𝚺subscript𝐱𝐮𝐮1𝐱1superscript𝝁T𝝁diag𝝁\bf\Sigma=-\frac{x_{{\bf u}{\bf u}}({\bf 1})}{x({\bf 1})}+\mbox{\boldmath{$\mu$}}^{\mathrm{T}}\mbox{\boldmath{$\mu$}}+{\rm diag}(\mbox{\boldmath{$\mu$}}),bold_Σ = - divide start_ARG bold_x start_POSTSUBSCRIPT bold_uu end_POSTSUBSCRIPT ( bold_1 ) end_ARG start_ARG bold_x ( bold_1 ) end_ARG + bold_italic_μ start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_italic_μ + roman_diag ( bold_italic_μ ) ,

where x=x(𝐮)𝑥𝑥𝐮x=x({\bf u})italic_x = italic_x ( bold_u ) (and 𝐲=𝐲(𝐮)=𝐲(x(𝐮),𝐮)𝐲𝐲𝐮𝐲𝑥𝐮𝐮{\bf y}={\bf y}({\bf u})={\bf y}(x({\bf u}),{\bf u})bold_y = bold_y ( bold_u ) = bold_y ( italic_x ( bold_u ) , bold_u )) is the solution of the (extended) system

(24) 𝐲𝐲\displaystyle{\bf y}bold_y =𝐅(x,𝐲,𝐮),absent𝐅𝑥𝐲𝐮\displaystyle={\bf F}(x,{\bf y},{\bf u}),= bold_F ( italic_x , bold_y , bold_u ) ,
(25) 00\displaystyle 0 =det(𝐈𝐅𝐲(x,𝐲,𝐮)).absent𝐈subscript𝐅𝐲𝑥𝐲𝐮\displaystyle=\det({\bf I}-{\bf F}_{\bf y}(x,{\bf y},{\bf u})).= roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) ) .

The proof of Theorem 2 is exactly the same as that given in [Drm97]. The main observation is that the assumptions above show that the solutions yj(x,𝐮)subscript𝑦𝑗𝑥𝐮y_{j}(x,{\bf u})italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_u ) admit a local representation of the form

yj(x,𝐮)=gj(x,𝐮)hj(x,𝐮)1xx(𝐮),subscript𝑦𝑗𝑥𝐮subscript𝑔𝑗𝑥𝐮subscript𝑗𝑥𝐮1𝑥𝑥𝐮y_{j}(x,{\bf u})=g_{j}(x,{\bf u})-h_{j}(x,{\bf u})\sqrt{1-\frac{x}{x({\bf u})}},italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_u ) = italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_u ) - italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_u ) square-root start_ARG 1 - divide start_ARG italic_x end_ARG start_ARG italic_x ( bold_u ) end_ARG end_ARG ,

(where 𝐮𝐮{\bf u}bold_u is close to 𝟏1{\bf 1}bold_1 and x𝑥xitalic_x close to x0=x(𝟏)subscript𝑥0𝑥1x_{0}=x({\bf 1})italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x ( bold_1 )). The assumption that the dependency graph is strongly connected ensures that the location of the singularity of all functions yj(x,𝐮)subscript𝑦𝑗𝑥𝐮y_{j}(x,{\bf u})italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , bold_u ) is determined by the common function x(𝐮)𝑥𝐮x({\bf u})italic_x ( bold_u ). Thus, we get the same property for G(x,𝐲(x,𝐮),𝐮)𝐺𝑥𝐲𝑥𝐮𝐮G(x,{\bf y}(x,{\bf u}),{\bf u})italic_G ( italic_x , bold_y ( italic_x , bold_u ) , bold_u ):

(26) G(x,𝐲(x,𝐮),𝐮)=g(x,𝐮)h(x,𝐮)1xx(𝐮)𝐺𝑥𝐲𝑥𝐮𝐮𝑔𝑥𝐮𝑥𝐮1𝑥𝑥𝐮G(x,{\bf y}(x,{\bf u}),{\bf u})=g(x,{\bf u})-h(x,{\bf u})\sqrt{1-\frac{x}{x({\bf u})}}italic_G ( italic_x , bold_y ( italic_x , bold_u ) , bold_u ) = italic_g ( italic_x , bold_u ) - italic_h ( italic_x , bold_u ) square-root start_ARG 1 - divide start_ARG italic_x end_ARG start_ARG italic_x ( bold_u ) end_ARG end_ARG

It is then well known (see [BR83, Drm94]) that a square-root singularity plus some minor conditions implies asymptotic normality of the coefficients (in the sense introduced above) with mean and covariance expressed in terms of derivatives of x(𝐮)𝑥𝐮x({\bf u})italic_x ( bold_u ). Note, for example, that the assumption dn,j>0subscript𝑑𝑛𝑗0d_{n,j}>0italic_d start_POSTSUBSCRIPT italic_n , italic_j end_POSTSUBSCRIPT > 0 for nn1𝑛subscript𝑛1n\geq n_{1}italic_n ≥ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ensures that cn>0subscript𝑐𝑛0c_{n}>0italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > 0 for sufficiently large n𝑛nitalic_n and from this follows that x0=x(𝟏)subscript𝑥0𝑥1x_{0}=x({\bf 1})italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x ( bold_1 ) is the only singularity on the radius of convergence of G(x,𝐲(x,𝟏),𝟏)𝐺𝑥𝐲𝑥11G(x,{\bf y}(x,{\bf 1}),{\bf 1})italic_G ( italic_x , bold_y ( italic_x , bold_1 ) , bold_1 ).

In what follows we comment on the evaluation of 𝝁𝝁\mubold_italic_μ and 𝚺𝚺\bf\Sigmabold_Σ. The problem is to extract the derivatives of x(𝐮)𝑥𝐮x({\bf u})italic_x ( bold_u ). The function x(𝐮)𝑥𝐮x({\bf u})italic_x ( bold_u ) is the solution of the system (2425) and is exactly the location of the singularity of the mapping x𝐲(x,𝐮)maps-to𝑥𝐲𝑥𝐮x\mapsto{\bf y}(x,{\bf u})italic_x ↦ bold_y ( italic_x , bold_u ) when 𝐮𝐮{\bf u}bold_u is fixed (and close to 𝟏1{\bf 1}bold_1).

Let x(𝐮)𝑥𝐮x({\bf u})italic_x ( bold_u ) and 𝐲(𝐮)=𝐲(x(𝐮),𝐮)𝐲𝐮𝐲𝑥𝐮𝐮{\bf y}({\bf u})={\bf y}(x({\bf u}),{\bf u})bold_y ( bold_u ) = bold_y ( italic_x ( bold_u ) , bold_u ) denote the solutions of (2425). Then we have

(27) 𝐲(𝐮)=𝐅(x(𝐮),𝐲(𝐮),𝐮).𝐲𝐮𝐅𝑥𝐮𝐲𝐮𝐮{\bf y}({\bf u})={\bf F}(x({\bf u}),{\bf y}({\bf u}),{\bf u}).bold_y ( bold_u ) = bold_F ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) .

Taking derivatives with respect to 𝐮𝐮{\bf u}bold_u we get

(28) 𝐲𝐮(𝐮)=𝐅x(x(𝐮),𝐲(𝐮),𝐮)x𝐮(𝐮)+𝐅𝐲(x(𝐮),𝐲(𝐮),𝐮)𝐲𝐮(𝐮)+𝐅𝐮(x(𝐮),𝐲(𝐮),𝐮),subscript𝐲𝐮𝐮subscript𝐅𝑥𝑥𝐮𝐲𝐮𝐮subscript𝑥𝐮𝐮subscript𝐅𝐲𝑥𝐮𝐲𝐮𝐮subscript𝐲𝐮𝐮subscript𝐅𝐮𝑥𝐮𝐲𝐮𝐮{\bf y}_{\bf u}({\bf u})={\bf F}_{x}(x({\bf u}),{\bf y}({\bf u}),{\bf u})x_{\bf u}({\bf u})+{\bf F}_{\bf y}(x({\bf u}),{\bf y}({\bf u}),{\bf u}){\bf y}_{\bf u}({\bf u})+{\bf F}_{\bf u}(x({\bf u}),{\bf y}({\bf u}),{\bf u}),bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( bold_u ) = bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( bold_u ) + bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( bold_u ) + bold_F start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) ,

where the three terms in 𝐅𝐅{\bf F}bold_F denote evaluations at (x(𝐮),𝐲(𝐮),𝐮)𝑥𝐮𝐲𝐮𝐮(x({\bf u}),{\bf y}({\bf u}),{\bf u})( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) of the partial derivatives of 𝐅𝐅{\bf F}bold_F, and where x𝐮subscript𝑥𝐮x_{\bf u}italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT and 𝐲𝐮subscript𝐲𝐮{\bf y}_{\bf u}bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT denote the Jacobian of x𝑥xitalic_x resp. 𝐲𝐲{\bf y}bold_y with respect to 𝐮𝐮{\bf u}bold_u. In particular, for 𝐮=𝟏𝐮1{\bf u}={\bf 1}bold_u = bold_1 we have x(𝟏)=x0𝑥1subscript𝑥0x({\bf 1})=x_{0}italic_x ( bold_1 ) = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and 𝐲(𝟏)=𝐲0𝐲1subscript𝐲0{\bf y}({\bf 1})={\bf y}_{0}bold_y ( bold_1 ) = bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and, of course

det(𝐈𝐅𝐲(x0,𝐲0,𝟏))=0.𝐈subscript𝐅𝐲subscript𝑥0subscript𝐲010\det({\bf I}-{\bf F}_{\bf y}(x_{0},{\bf y}_{0},{\bf 1}))=0.roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) ) = 0 .

Since 𝐅𝐲subscript𝐅𝐲{\bf F}_{\bf y}bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT is a non-negative matrix and the dependency graph is strongly connected there is a unique Perron-Frobenius eigenvalue of multiplicity 1. Here this eigenvalue equals 1. Thus, 𝐈𝐅𝐲𝐈subscript𝐅𝐲{\bf I}-{\bf F}_{\bf y}bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT has rank N1𝑁1N-1italic_N - 1 and has (up to scaling) a unique positive left eigenvector 𝐛Tsuperscript𝐛T{\bf b}^{\mathrm{T}}bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT:

𝐛T(𝐈𝐅𝐲(x0,𝐲0,𝟏))=𝟎.superscript𝐛T𝐈subscript𝐅𝐲subscript𝑥0subscript𝐲010{\bf b}^{\mathrm{T}}({\bf I}-{\bf F}_{\bf y}(x_{0},{\bf y}_{0},{\bf 1}))={\bf 0}.bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT ( bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) ) = bold_0 .

From (28) we obtain

(𝐈𝐅𝐲(x0,𝐲0,𝟏))𝐲𝐮(𝟏)=𝐅x(x0,𝐲0,𝟏)x𝐮(𝟏)+𝐅𝐮(x0,𝐲0,𝟏).𝐈subscript𝐅𝐲subscript𝑥0subscript𝐲01subscript𝐲𝐮1subscript𝐅𝑥subscript𝑥0subscript𝐲01subscript𝑥𝐮1subscript𝐅𝐮subscript𝑥0subscript𝐲01({\bf I}-{\bf F}_{\bf y}(x_{0},{\bf y}_{0},{\bf 1})){\bf y}_{\bf u}({\bf 1})={\bf F}_{x}(x_{0},{\bf y}_{0},{\bf 1})x_{\bf u}({\bf 1})+{\bf F}_{\bf u}(x_{0},{\bf y}_{0},{\bf 1}).( bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) ) bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( bold_1 ) = bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( bold_1 ) + bold_F start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) .

By multiplying 𝐛Tsuperscript𝐛T{\bf b}^{\mathrm{T}}bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT from the left we thus get

(29) 𝐛T𝐅x(x0,𝐲0,𝟏)x𝐮+𝐛T𝐅𝐮(x0,𝐲0,𝟏)=0superscript𝐛Tsubscript𝐅𝑥subscript𝑥0subscript𝐲01subscript𝑥𝐮superscript𝐛Tsubscript𝐅𝐮subscript𝑥0subscript𝐲010{\bf b}^{\mathrm{T}}{\bf F}_{x}(x_{0},{\bf y}_{0},{\bf 1})x_{\bf u}+{\bf b}^{\mathrm{T}}{\bf F}_{\bf u}(x_{0},{\bf y}_{0},{\bf 1})=0bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT + bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) = 0

and consequently

𝝁=1x0𝐛T𝐅𝐮(x0,𝐲0,𝟏)𝐛T𝐅x(x0,𝐲0,𝟏)𝝁1subscript𝑥0superscript𝐛Tsubscript𝐅𝐮subscript𝑥0subscript𝐲01superscript𝐛Tsubscript𝐅𝑥subscript𝑥0subscript𝐲01\mbox{\boldmath{$\mu$}}=\frac{1}{x_{0}}\frac{{\bf b}^{\mathrm{T}}{\bf F}_{\bf u}(x_{0},{\bf y}_{0},{\bf 1})}{{\bf b}^{\mathrm{T}}{\bf F}_{x}(x_{0},{\bf y}_{0},{\bf 1})}bold_italic_μ = divide start_ARG 1 end_ARG start_ARG italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG divide start_ARG bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) end_ARG start_ARG bold_b start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_1 ) end_ARG

The derivation of 𝚺𝚺{\bf\Sigma}bold_Σ is more involved. We first define 𝐛(x,𝐲,𝐮)𝐛𝑥𝐲𝐮{\bf b}(x,{\bf y},{\bf u})bold_b ( italic_x , bold_y , bold_u ) as the (generalized) vector product666More precisely this is the wedge product combined with the Hodge duality. of the N1𝑁1N-1italic_N - 1 last columns of the matrix 𝐈𝐅𝐲(x,𝐲,𝐮)𝐈subscript𝐅𝐲𝑥𝐲𝐮{\bf I}-{\bf F}_{\bf y}(x,{\bf y},{\bf u})bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ). Observe that

D(x,𝐲,𝐮):=(𝐛T(x,𝐲,𝐮)(𝐈𝐅𝐲(x,𝐲,𝐮)))1=det(𝐈𝐅𝐲(x,𝐲,𝐮)).assign𝐷𝑥𝐲𝐮subscriptsuperscript𝐛𝑇𝑥𝐲𝐮𝐈subscript𝐅𝐲𝑥𝐲𝐮1𝐈subscript𝐅𝐲𝑥𝐲𝐮D(x,{\bf y},{\bf u}):=\left({\bf b}^{T}(x,{\bf y},{\bf u})\left({\bf I}-{\bf F}_{\bf y}(x,{\bf y},{\bf u})\right)\right)_{1}=\det\left({\bf I}-{\bf F}_{\bf y}(x,{\bf y},{\bf u})\right).italic_D ( italic_x , bold_y , bold_u ) := ( bold_b start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_x , bold_y , bold_u ) ( bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) ) ) start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ( italic_x , bold_y , bold_u ) ) .

In particular we have

D(x(𝐮),𝐲(𝐮),𝐮)=0.𝐷𝑥𝐮𝐲𝐮𝐮0D(x({\bf u}),{\bf y}({\bf u}),{\bf u})=0.italic_D ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) = 0 .

Then from

(𝐈𝐅𝐲)𝐲𝐮𝐈subscript𝐅𝐲subscript𝐲𝐮\displaystyle({\bf I}-{\bf F}_{\bf y}){\bf y}_{\bf u}( bold_I - bold_F start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ) bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT =𝐅xx𝐮+𝐅𝐮,absentsubscript𝐅𝑥subscript𝑥𝐮subscript𝐅𝐮\displaystyle={\bf F}_{x}x_{\bf u}+{\bf F}_{\bf u},= bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT + bold_F start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ,
(30) D𝐲𝐲𝐮subscript𝐷𝐲subscript𝐲𝐮\displaystyle-D_{\bf y}{\bf y}_{\bf u}- italic_D start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT =Dxx𝐮+D𝐮absentsubscript𝐷𝑥subscript𝑥𝐮subscript𝐷𝐮\displaystyle=D_{x}x_{\bf u}+D_{\bf u}= italic_D start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT

we can calculate 𝐲𝐮subscript𝐲𝐮{\bf y}_{\bf u}bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT. (The first system has rank N1𝑁1N-1italic_N - 1, this means that we can skip the first equation. This reduced system is then completed to a regular system by appending the second equation (30).)

We now set

d1(𝐮)subscript𝑑1𝐮\displaystyle d_{1}({\bf u})italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( bold_u ) =d1(x(𝐮),𝐲(𝐮),𝐮)=𝐛(x(𝐮),𝐲(𝐮),𝐮)T𝐅x(x(𝐮),𝐲(𝐮),𝐮)absentsubscript𝑑1𝑥𝐮𝐲𝐮𝐮𝐛superscript𝑥𝐮𝐲𝐮𝐮Tsubscript𝐅𝑥𝑥𝐮𝐲𝐮𝐮\displaystyle=d_{1}(x({\bf u}),{\bf y}({\bf u}),{\bf u})={\bf b}(x({\bf u}),{\bf y}({\bf u}),{\bf u})^{\mathrm{T}}{\bf F}_{x}(x({\bf u}),{\bf y}({\bf u}),{\bf u})= italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) = bold_b ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u )
𝐝2(𝐮)subscript𝐝2𝐮\displaystyle{\bf d}_{2}({\bf u})bold_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( bold_u ) =𝐝2(x(𝐮),𝐲(𝐮),𝐮)=𝐛(x(𝐮),𝐲(𝐮),𝐮)T𝐅𝐮(x(𝐮),𝐲(𝐮),𝐮).absentsubscript𝐝2𝑥𝐮𝐲𝐮𝐮𝐛superscript𝑥𝐮𝐲𝐮𝐮Tsubscript𝐅𝐮𝑥𝐮𝐲𝐮𝐮\displaystyle={\bf d}_{2}(x({\bf u}),{\bf y}({\bf u}),{\bf u})={\bf b}(x({\bf u}),{\bf y}({\bf u}),{\bf u})^{\mathrm{T}}{\bf F}_{\bf u}(x({\bf u}),{\bf y}({\bf u}),{\bf u}).= bold_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) = bold_b ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT bold_F start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( italic_x ( bold_u ) , bold_y ( bold_u ) , bold_u ) .

By differentiating equation (29) we get

(31) x𝐮𝐮(𝐮)=(d1xx𝐮+d1𝐲𝐲𝐮+d1𝐮)x𝐮+(𝐝2xx𝐮+𝐝2𝐲𝐲𝐮+𝐝2𝐮)d1,subscript𝑥𝐮𝐮𝐮subscript𝑑1𝑥subscript𝑥𝐮subscript𝑑1𝐲subscript𝐲𝐮subscript𝑑1𝐮subscript𝑥𝐮subscript𝐝2𝑥subscript𝑥𝐮subscript𝐝2𝐲subscript𝐲𝐮subscript𝐝2𝐮subscript𝑑1x_{{\bf u}{\bf u}}({\bf u})=-\frac{(d_{1x}x_{\bf u}+d_{1{\bf y}}{\bf y}_{\bf u}+d_{1{\bf u}})x_{\bf u}+({\bf d}_{2x}x_{\bf u}+{\bf d}_{2{\bf y}}{\bf y}_{\bf u}+{\bf d}_{2{\bf u}})}{d_{1}},italic_x start_POSTSUBSCRIPT bold_uu end_POSTSUBSCRIPT ( bold_u ) = - divide start_ARG ( italic_d start_POSTSUBSCRIPT 1 italic_x end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT + italic_d start_POSTSUBSCRIPT 1 bold_y end_POSTSUBSCRIPT bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT + italic_d start_POSTSUBSCRIPT 1 bold_u end_POSTSUBSCRIPT ) italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT + ( bold_d start_POSTSUBSCRIPT 2 italic_x end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT + bold_d start_POSTSUBSCRIPT 2 bold_y end_POSTSUBSCRIPT bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT + bold_d start_POSTSUBSCRIPT 2 bold_u end_POSTSUBSCRIPT ) end_ARG start_ARG italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ,

where d1x,d1𝐲,d1𝐮,𝐝2x,𝐝2𝐲,𝐝2𝐮subscript𝑑1𝑥subscript𝑑1𝐲subscript𝑑1𝐮subscript𝐝2𝑥subscript𝐝2𝐲subscript𝐝2𝐮d_{1x},d_{1{\bf y}},d_{1{\bf u}},{\bf d}_{2x},{\bf d}_{2{\bf y}},{\bf d}_{2{\bf u}}italic_d start_POSTSUBSCRIPT 1 italic_x end_POSTSUBSCRIPT , italic_d start_POSTSUBSCRIPT 1 bold_y end_POSTSUBSCRIPT , italic_d start_POSTSUBSCRIPT 1 bold_u end_POSTSUBSCRIPT , bold_d start_POSTSUBSCRIPT 2 italic_x end_POSTSUBSCRIPT , bold_d start_POSTSUBSCRIPT 2 bold_y end_POSTSUBSCRIPT , bold_d start_POSTSUBSCRIPT 2 bold_u end_POSTSUBSCRIPT denote the respective partial derivatives and where we omitted the dependence on 𝐮𝐮{\bf u}bold_u. With the knowledge of x0,𝐲0subscript𝑥0subscript𝐲0x_{0},{\bf y}_{0}italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and 𝐲𝐮(𝟏)subscript𝐲𝐮1{\bf y}_{\bf u}({\bf 1})bold_y start_POSTSUBSCRIPT bold_u end_POSTSUBSCRIPT ( bold_1 ) we can now evaluate x𝐮𝐮subscript𝑥𝐮𝐮x_{{\bf u}{\bf u}}italic_x start_POSTSUBSCRIPT bold_uu end_POSTSUBSCRIPT at 𝐮=𝟏𝐮1{\bf u}={\bf 1}bold_u = bold_1 and we finally calculate 𝚺𝚺\bf\Sigmabold_Σ from (23).

Appendix C Proof of Lemma 1

In this appendix we will prove Lemma 1 saying that the determinant det(𝐈𝐅𝐚(x,𝐚,1))𝐈subscript𝐅𝐚𝑥𝐚1\det\left({\bf I}-{\bf F}_{\bf a}(x,{\bf a},1)\right)roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x , bold_a , 1 ) ) is given by

det(𝐈𝐅𝐚(x,𝐚,1))=1xea0+a1++aL.𝐈subscript𝐅𝐚𝑥𝐚11𝑥superscript𝑒subscript𝑎0subscript𝑎1subscript𝑎𝐿\det\left({\bf I}-{\bf F}_{\bf a}(x,{\bf a},1)\right)=1-xe^{a_{0}+a_{1}+\cdots+a_{L}}.roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x , bold_a , 1 ) ) = 1 - italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT .

We first observe that the sum of all rows of 𝐈𝐅𝐚(x,𝐚,1)𝐈subscript𝐅𝐚𝑥𝐚1{\bf I}-{\bf F}_{\bf a}(x,{\bf a},1)bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x , bold_a , 1 ) equals

(1xea0+a1++aL,1xea0+a1++aL,,1xea0+a1++aL),1𝑥superscript𝑒subscript𝑎0subscript𝑎1subscript𝑎𝐿1𝑥superscript𝑒subscript𝑎0subscript𝑎1subscript𝑎𝐿1𝑥superscript𝑒subscript𝑎0subscript𝑎1subscript𝑎𝐿\left(1-xe^{a_{0}+a_{1}+\cdots+a_{L}},1-xe^{a_{0}+a_{1}+\cdots+a_{L}},\ldots,1-xe^{a_{0}+a_{1}+\cdots+a_{L}}\right),( 1 - italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , 1 - italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , … , 1 - italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) ,

compare with (14). Hence, we get

det(𝐈𝐅𝐚(x,𝐚,1))=(1xea0+a1++aL)det𝐌(x,𝐚),𝐈subscript𝐅𝐚𝑥𝐚11𝑥superscript𝑒subscript𝑎0subscript𝑎1subscript𝑎𝐿𝐌𝑥𝐚\det\left({\bf I}-{\bf F}_{\bf a}(x,{\bf a},1)\right)=(1-xe^{a_{0}+a_{1}+\ldots+a_{L}})\det{\bf M}(x,{\bf a}),roman_det ( bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ( italic_x , bold_a , 1 ) ) = ( 1 - italic_x italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + … + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) roman_det bold_M ( italic_x , bold_a ) ,

where 𝐌(x,𝐚)𝐌𝑥𝐚{\bf M}(x,{\bf a})bold_M ( italic_x , bold_a ) denotes the matrix 𝐈𝐅𝐚𝐈subscript𝐅𝐚{\bf I}-{\bf F}_{\bf a}bold_I - bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT where we replace the first row by (1,1,,1)111(1,1,\ldots,1)( 1 , 1 , … , 1 ). Thus, it remains to prove that det𝐌(x,𝐚)=1𝐌𝑥𝐚1\det{\bf M}(x,{\bf a})=1roman_det bold_M ( italic_x , bold_a ) = 1.

For this purpose we have to be more explicit with the partition 𝒜={a0,a1,,aL}𝒜subscript𝑎0subscript𝑎1subscript𝑎𝐿\mathcal{A}=\{a_{0},a_{1},\dots,a_{L}\}caligraphic_A = { italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT }. More precisely we construct 𝒜𝒜\mathcal{A}caligraphic_A recursively from level to level. This procedure is similar to that of Proposition 3 but not the same. In order to make our arguments more transparent we restrict ourselves to 4 steps. Note that this procedure also provides a recursive description of the polynomials Pj(𝐚,1)subscript𝑃𝑗𝐚1P_{j}({\bf a},1)italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( bold_a , 1 ).

One starts with 𝒜0={d0,d1}subscript𝒜0subscript𝑑0subscript𝑑1\mathcal{A}_{0}=\{d_{0},d_{1}\}caligraphic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }, where d0=a0subscript𝑑0subscript𝑎0d_{0}=a_{0}italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and d1=pa0subscript𝑑1𝑝subscript𝑎0d_{1}=p\setminus a_{0}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_p ∖ italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. This means that d0subscript𝑑0d_{0}italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT collects all trees where the root out-degree is not contained in D𝐷Ditalic_D and d1subscript𝑑1d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT those where it is contained in D𝐷Ditalic_D. For example, if D={2}𝐷2D=\{2\}italic_D = { 2 } then the generating functions of this (trivial) partition are given by d1(x,1)=xp(x)2/2subscript𝑑1𝑥1𝑥𝑝superscript𝑥22d_{1}(x,1)=xp(x)^{2}/2italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_x italic_p ( italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 and by d0(x,1)=p(x)d1(x,1)=p(x)xp(x)2/2subscript𝑑0𝑥1𝑝𝑥subscript𝑑1𝑥1𝑝𝑥𝑥𝑝superscript𝑥22d_{0}(x,1)=p(x)-d_{1}(x,1)=p(x)-xp(x)^{2}/2italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_p ( italic_x ) - italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_p ( italic_x ) - italic_x italic_p ( italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2.

Then we partition d1subscript𝑑1d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT according to structure of the subtrees of the root, where we distinguish between the previous classes d0subscript𝑑0d_{0}italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and d1subscript𝑑1d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. We get 𝒜1={c0,c1,,cm}subscript𝒜1subscript𝑐0subscript𝑐1subscript𝑐𝑚\mathcal{A}_{1}=\{c_{0},c_{1},\ldots,c_{m}\}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = { italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }, where c0=d0subscript𝑐0subscript𝑑0c_{0}=d_{0}italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and c1cm=d1direct-sumsubscript𝑐1subscript𝑐𝑚subscript𝑑1c_{1}\oplus\ldots\oplus c_{m}=d_{1}italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ … ⊕ italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. In particular, if D={2}𝐷2D=\{2\}italic_D = { 2 } then m=3𝑚3m=3italic_m = 3, the class c1subscript𝑐1c_{1}italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT collects all trees with root out-degree 2222 where both subtrees of the root are in class a0=d0subscript𝑎0subscript𝑑0a_{0}=d_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, c2subscript𝑐2c_{2}italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT collects all trees with with root out-degree 2222 where one subtree of the root is in class a0=d0subscript𝑎0subscript𝑑0a_{0}=d_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and the other one in class d1subscript𝑑1d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, and c3subscript𝑐3c_{3}italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT collects those trees where both subtrees of the root are in class d1subscript𝑑1d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The corresponding generating functions are given by c1(x,1)=xd0(x,1)2/2subscript𝑐1𝑥1𝑥subscript𝑑0superscript𝑥122c_{1}(x,1)=xd_{0}(x,1)^{2}/2italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_x italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2, by c2(x,1)=xd0(x,1)d1(x,1)subscript𝑐2𝑥1𝑥subscript𝑑0𝑥1subscript𝑑1𝑥1c_{2}(x,1)=xd_{0}(x,1)d_{1}(x,1)italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_x italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ), and by c3(x,1)=xd1(x,1)2/2subscript𝑐3𝑥1𝑥subscript𝑑1superscript𝑥122c_{3}(x,1)=xd_{1}(x,1)^{2}/2italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_x italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2. Of course, we also have c0(x,1)=d0(x,1)subscript𝑐0𝑥1subscript𝑑0𝑥1c_{0}(x,1)=d_{0}(x,1)italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) and c1(x,1)+c2(x,1)+c3(x,1)=d1(x,1)subscript𝑐1𝑥1subscript𝑐2𝑥1subscript𝑐3𝑥1subscript𝑑1𝑥1c_{1}(x,1)+c_{2}(x,1)+c_{3}(x,1)=d_{1}(x,1)italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x , 1 ) + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ).

In the same fashion we proceed further. We partition cssubscript𝑐𝑠c_{s}italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT (1sm1𝑠𝑚1\leq s\leq m1 ≤ italic_s ≤ italic_m) according to the structure of the subtrees of the root (that are now taken from {c1,,cm}subscript𝑐1subscript𝑐𝑚\{c_{1},\ldots,c_{m}\}{ italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }) and denote them by 𝒜2={b0,b1,,b}subscript𝒜2subscript𝑏0subscript𝑏1subscript𝑏\mathcal{A}_{2}=\{b_{0},b_{1},\ldots,b_{\ell}\}caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = { italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_b start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT }. Further we define sets Cssubscript𝐶𝑠C_{s}italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT by cs=rCsbrsubscript𝑐𝑠subscriptdirect-sum𝑟subscript𝐶𝑠subscript𝑏𝑟c_{s}=\bigoplus_{r\in C_{s}}b_{r}italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT italic_r ∈ italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT. If D={2}𝐷2D=\{2\}italic_D = { 2 } then b0=c0subscript𝑏0subscript𝑐0b_{0}=c_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, b1=c1subscript𝑏1subscript𝑐1b_{1}=c_{1}italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, c2subscript𝑐2c_{2}italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is divided into three parts, and c3subscript𝑐3c_{3}italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT is divided into 6 parts: C1={1}subscript𝐶11C_{1}=\{1\}italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = { 1 }, C2={2,3,4}subscript𝐶2234C_{2}=\{2,3,4\}italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = { 2 , 3 , 4 }, C3={5,6,7,8,9,10}subscript𝐶35678910C_{3}=\{5,6,7,8,9,10\}italic_C start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = { 5 , 6 , 7 , 8 , 9 , 10 }.777By the way this leads to the partition that is used in the proof of Theorem 1 resp. of Proposition 1.

Finally, we partition bjsubscript𝑏𝑗b_{j}italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT (j1𝑗1j\geq 1italic_j ≥ 1) according according to the structure of the subtrees of the root that are taken from the bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and denote them by 𝒜={a0,a1,,aL}𝒜subscript𝑎0subscript𝑎1subscript𝑎𝐿\mathcal{A}=\{a_{0},a_{1},\ldots,a_{L}\}caligraphic_A = { italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT }. As in the previous step we define sets Brsubscript𝐵𝑟B_{r}italic_B start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT by br=jBrajsubscript𝑏𝑟subscriptdirect-sum𝑗subscript𝐵𝑟subscript𝑎𝑗b_{r}=\bigoplus_{j\in B_{r}}a_{j}italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT italic_j ∈ italic_B start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT. In general we have to iterate this procedure until a certain level and get almost the same partition as in the proof of Proposition 1. The only difference is that at the lowest level we only distinguish between nodes with degree in D𝐷Ditalic_D and degree not in D𝐷Ditalic_D. However this is no real restriction as we can extend the partition above with an additional level and we will have a well-defined number of additional occurrences for each class. We again obtain a partition which fits Proposition 1.

We recall that this recursive procedure directly provides a recursive description of the system of functional equations. In particular we have

aj(x,1)=xPj(a0(x,1),a1(x,1),,aL(x,1),1),subscript𝑎𝑗𝑥1𝑥subscript𝑃𝑗subscript𝑎0𝑥1subscript𝑎1𝑥1subscript𝑎𝐿𝑥11a_{j}(x,1)=x\,P_{j}(a_{0}(x,1),a_{1}(x,1),\ldots,a_{L}(x,1),1),italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_x italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) , … , italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_x , 1 ) , 1 ) ,

where Pj(,1)subscript𝑃𝑗1P_{j}(\cdot,1)italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( ⋅ , 1 ) can be actually written as a polynomial in b0,b1,,bsubscript𝑏0subscript𝑏1subscript𝑏b_{0},b_{1},\ldots,b_{\ell}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_b start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT.

Next

br(x,1)=xQr(b0(x,1),b1(x,1),,b(x,1),1),subscript𝑏𝑟𝑥1𝑥subscript𝑄𝑟subscript𝑏0𝑥1subscript𝑏1𝑥1subscript𝑏𝑥11b_{r}(x,1)=x\,Q_{r}(b_{0}(x,1),b_{1}(x,1),\ldots,b_{\ell}(x,1),1),italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_x italic_Q start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) , italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) , … , italic_b start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_x , 1 ) , 1 ) ,

where Qr(,1)subscript𝑄𝑟1Q_{r}(\cdot,1)italic_Q start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( ⋅ , 1 ) can be written as a polynomial in c0,c1,,cmsubscript𝑐0subscript𝑐1subscript𝑐𝑚c_{0},c_{1},\ldots,c_{m}italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. Further,

Qr=jBrPj.subscript𝑄𝑟subscript𝑗subscript𝐵𝑟subscript𝑃𝑗Q_{r}=\sum_{j\in B_{r}}P_{j}.italic_Q start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j ∈ italic_B start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT .

In other words, the sum jBrPjsubscript𝑗subscript𝐵𝑟subscript𝑃𝑗\sum_{j\in B_{r}}P_{j}∑ start_POSTSUBSCRIPT italic_j ∈ italic_B start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT can be written as polynomial in crsubscript𝑐𝑟c_{r}italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT.

Finally,

cs(x,1)=xRs(c0(x,1),c1(x,1),,cm(x,1)),subscript𝑐𝑠𝑥1𝑥subscript𝑅𝑠subscript𝑐0𝑥1subscript𝑐1𝑥1subscript𝑐𝑚𝑥1c_{s}(x,1)=x\,R_{s}(c_{0}(x,1),c_{1}(x,1),\ldots,c_{m}(x,1)),italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_x , 1 ) = italic_x italic_R start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x , 1 ) , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , 1 ) , … , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_x , 1 ) ) ,

where Rs(,1)subscript𝑅𝑠1R_{s}(\cdot,1)italic_R start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( ⋅ , 1 ) can be written as a polynomial in d0=a0subscript𝑑0subscript𝑎0d_{0}=a_{0}italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and d1=a1++aLsubscript𝑑1subscript𝑎1subscript𝑎𝐿d_{1}=a_{1}+\cdots+a_{L}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT and we have

Rs=rCsQr.subscript𝑅𝑠subscript𝑟subscript𝐶𝑠subscript𝑄𝑟R_{s}=\sum_{r\in C_{s}}Q_{r}.italic_R start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_r ∈ italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT .

Let 𝐆(x,𝐚)𝐆𝑥𝐚{\bf G}(x,{\bf a})bold_G ( italic_x , bold_a ) denote the L×L𝐿𝐿L\times Litalic_L × italic_L-submatrix of 𝐅𝐚subscript𝐅𝐚{\bf F}_{\bf a}bold_F start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT where we omit the first row and column. Then 𝐆(x,𝐚)𝐆𝑥𝐚{\bf G}(x,{\bf a})bold_G ( italic_x , bold_a ) has the following structure:

𝐆(x,𝐚)=(G11G1mGm1Gmm),𝐆𝑥𝐚subscript𝐺11subscript𝐺1𝑚missing-subexpressionsubscript𝐺𝑚1subscript𝐺𝑚𝑚{\bf G}(x,{\bf a})=\left(\begin{array}[]{ccc}G_{11}&\cdots&G_{1m}\\ \vdots&&\vdots\\ G_{m1}&\cdots&G_{mm}\end{array}\right),bold_G ( italic_x , bold_a ) = ( start_ARRAY start_ROW start_CELL italic_G start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_G start_POSTSUBSCRIPT 1 italic_m end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_G start_POSTSUBSCRIPT italic_m 1 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_G start_POSTSUBSCRIPT italic_m italic_m end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ,

where

Gss′′=(Brr′′)rCs,r′′Cs′′subscript𝐺superscript𝑠superscript𝑠′′subscriptsubscript𝐵superscript𝑟superscript𝑟′′formulae-sequencesuperscript𝑟subscript𝐶superscript𝑠superscript𝑟′′subscript𝐶superscript𝑠′′G_{s^{\prime}s^{\prime\prime}}=\left(\begin{array}[]{c}B_{r^{\prime}r^{\prime\prime}}\end{array}\right)_{r^{\prime}\in C_{s^{\prime}},r^{\prime\prime}\in C_{s^{\prime\prime}}}italic_G start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT

and

Brr′′=(xPi,aj)iBr,jBr′′.subscript𝐵superscript𝑟superscript𝑟′′subscript𝑥subscript𝑃𝑖subscript𝑎𝑗formulae-sequence𝑖subscript𝐵superscript𝑟𝑗subscript𝐵superscript𝑟′′B_{r^{\prime}r^{\prime\prime}}=\left(\begin{array}[]{c}xP_{i,a_{j}}\end{array}\right)_{i\in B_{r^{\prime}},j\in B_{r^{\prime\prime}}}.italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL italic_x italic_P start_POSTSUBSCRIPT italic_i , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) start_POSTSUBSCRIPT italic_i ∈ italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_j ∈ italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT .

The condition that Pisubscript𝑃𝑖P_{i}italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT can be written as a polynomial in bjsubscript𝑏𝑗b_{j}italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT implies that Pi,aj1=Pi,aj2subscript𝑃𝑖subscript𝑎subscript𝑗1subscript𝑃𝑖subscript𝑎subscript𝑗2P_{i,a_{j_{1}}}=P_{i,a_{j_{2}}}italic_P start_POSTSUBSCRIPT italic_i , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_P start_POSTSUBSCRIPT italic_i , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT for all j1,j2Br′′subscript𝑗1subscript𝑗2subscript𝐵superscript𝑟′′j_{1},j_{2}\in B_{r^{\prime\prime}}italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, that is, each row of Brr′′subscript𝐵superscript𝑟superscript𝑟′′B_{r^{\prime}r^{\prime\prime}}italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is either zero or all entries are the same.

Further, if we fix rsuperscript𝑟r^{\prime}italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and sum over all rows iBr𝑖subscript𝐵superscript𝑟i\in B_{r^{\prime}}italic_i ∈ italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT then we get

iBrxPi,aj=xQr,aj.subscript𝑖subscript𝐵superscript𝑟𝑥subscript𝑃𝑖subscript𝑎𝑗𝑥subscript𝑄superscript𝑟subscript𝑎𝑗\sum_{i\in B_{r^{\prime}}}xP_{i,a_{j}}=xQ_{r^{\prime},a_{j}}.∑ start_POSTSUBSCRIPT italic_i ∈ italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_x italic_P start_POSTSUBSCRIPT italic_i , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_x italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT .

Since Qrsubscript𝑄superscript𝑟Q_{r^{\prime}}italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT can be written as a polynomial in cssubscript𝑐𝑠c_{s}italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT (0sm0𝑠𝑚0\leq s\leq m0 ≤ italic_s ≤ italic_m) we have Qr,aj1=Qr,aj2subscript𝑄superscript𝑟subscript𝑎subscript𝑗1subscript𝑄superscript𝑟subscript𝑎subscript𝑗2Q_{r^{\prime},a_{j_{1}}}=Q_{r^{\prime},a_{j_{2}}}italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT for all j1,j2C¯s′′subscript𝑗1subscript𝑗2subscript¯𝐶superscript𝑠′′j_{1},j_{2}\in\bar{C}_{s^{\prime\prime}}italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, where we set C¯s=rCsBrsubscript¯𝐶𝑠subscript𝑟subscript𝐶𝑠subscript𝐵𝑟\bar{C}_{s}=\bigcup\limits_{r\in C_{s}}B_{r}over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = ⋃ start_POSTSUBSCRIPT italic_r ∈ italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT.

Similarly if we fix ssuperscript𝑠s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and sum over all rows iC¯s𝑖subscript¯𝐶superscript𝑠i\in\bar{C}_{s^{\prime}}italic_i ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT then we get

iC¯sxPi,aj=xRs,aj.subscript𝑖subscript¯𝐶superscript𝑠𝑥subscript𝑃𝑖subscript𝑎𝑗𝑥subscript𝑅superscript𝑠subscript𝑎𝑗\sum_{i\in\bar{C}_{s^{\prime}}}xP_{i,a_{j}}=xR_{s^{\prime},a_{j}}.∑ start_POSTSUBSCRIPT italic_i ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_x italic_P start_POSTSUBSCRIPT italic_i , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_x italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT .

Since Rssubscript𝑅superscript𝑠R_{s^{\prime}}italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT can be written as a polynomial in d0=a0subscript𝑑0subscript𝑎0d_{0}=a_{0}italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and d1=a1++aLsubscript𝑑1subscript𝑎1subscript𝑎𝐿d_{1}=a_{1}+\cdots+a_{L}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_a start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT we have Rs,aj1=Rs,aj2subscript𝑅superscript𝑠subscript𝑎subscript𝑗1subscript𝑅superscript𝑠subscript𝑎subscript𝑗2R_{s^{\prime},a_{j_{1}}}=R_{s^{\prime},a_{j_{2}}}italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT for all 1j1,j2Lformulae-sequence1subscript𝑗1subscript𝑗2𝐿1\leq j_{1},j_{2}\leq L1 ≤ italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ italic_L.

Now we will calculate the determinant of the matrix

𝐌(x,𝐚)𝐌𝑥𝐚\displaystyle{\bf M}(x,{\bf a})bold_M ( italic_x , bold_a ) =(1110𝐈𝟎0𝟎𝐈)(0𝟎𝟎×G11G1m×Gm1Gmm)absent1110𝐈000𝐈000subscript𝐺11subscript𝐺1𝑚missing-subexpressionsubscript𝐺𝑚1subscript𝐺𝑚𝑚\displaystyle=\left(\begin{array}[]{cccc}1&1\cdots&\cdots&\cdots 1\\ 0&{\bf I}&\cdots&{\bf 0}\\ \vdots&\vdots&\ddots&\vdots\\ 0&{\bf 0}&\cdots&{\bf I}\end{array}\right)-\left(\begin{array}[]{cccc}0&{\bf 0}&\cdots&{\bf 0}\\ \times&G_{11}&\cdots&G_{1m}\\ \vdots&\vdots&&\vdots\\ \times&G_{m1}&\cdots&G_{mm}\end{array}\right)= ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 ⋯ end_CELL start_CELL ⋯ end_CELL start_CELL ⋯ 1 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL bold_I end_CELL start_CELL ⋯ end_CELL start_CELL bold_0 end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL bold_0 end_CELL start_CELL ⋯ end_CELL start_CELL bold_I end_CELL end_ROW end_ARRAY ) - ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL bold_0 end_CELL start_CELL ⋯ end_CELL start_CELL bold_0 end_CELL end_ROW start_ROW start_CELL × end_CELL start_CELL italic_G start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_G start_POSTSUBSCRIPT 1 italic_m end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL × end_CELL start_CELL italic_G start_POSTSUBSCRIPT italic_m 1 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_G start_POSTSUBSCRIPT italic_m italic_m end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY )
=(111×𝐈G11G1m×Gm1𝐈Gmm).absent111𝐈subscript𝐺11subscript𝐺1𝑚missing-subexpressionsubscript𝐺𝑚1𝐈subscript𝐺𝑚𝑚\displaystyle=\left(\begin{array}[]{cccc}1&1\cdots&\cdots&\cdots 1\\ \times&{\bf I}-G_{11}&\cdots&-G_{1m}\\ \vdots&\vdots&&\vdots\\ \times&-G_{m1}&\cdots&{\bf I}-G_{mm}\end{array}\right).= ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 ⋯ end_CELL start_CELL ⋯ end_CELL start_CELL ⋯ 1 end_CELL end_ROW start_ROW start_CELL × end_CELL start_CELL bold_I - italic_G start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL - italic_G start_POSTSUBSCRIPT 1 italic_m end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL × end_CELL start_CELL - italic_G start_POSTSUBSCRIPT italic_m 1 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL bold_I - italic_G start_POSTSUBSCRIPT italic_m italic_m end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) .

(By ×\times× we denote an entry we do not care.) We now perform the following row operations. For every s=1,,msuperscript𝑠1𝑚s^{\prime}=1,\ldots,mitalic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 1 , … , italic_m we substitute the first row of

(×Gs1𝐈GssGsm)subscript𝐺superscript𝑠1𝐈subscript𝐺superscript𝑠superscript𝑠subscript𝐺superscript𝑠𝑚missing-subexpression\left(\begin{array}[]{cccccc}\times&-G_{s^{\prime}1}&\cdots&{\bf I}-G_{s^{\prime}s^{\prime}}&\cdots-G_{s^{\prime}m}\end{array}\right)( start_ARRAY start_ROW start_CELL × end_CELL start_CELL - italic_G start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL bold_I - italic_G start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL ⋯ - italic_G start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_m end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW end_ARRAY )

by the sum of the corresponding rows iC¯s𝑖subscript¯𝐶superscript𝑠i\in\bar{C}_{s^{\prime}}italic_i ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Since Rs,aj1=Rs,aj2subscript𝑅superscript𝑠subscript𝑎subscript𝑗1subscript𝑅superscript𝑠subscript𝑎subscript𝑗2R_{s^{\prime},a_{j_{1}}}=R_{s^{\prime},a_{j_{2}}}italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT for all 1j1,j2Lformulae-sequence1subscript𝑗1subscript𝑗2𝐿1\leq j_{1},j_{2}\leq L1 ≤ italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ italic_L this sum of the rows has the form

(×xRs,axRs,a1xRs,a1xRs,axRs,axRs,a)𝑥subscript𝑅superscript𝑠𝑎𝑥subscript𝑅superscript𝑠𝑎1𝑥subscript𝑅superscript𝑠𝑎1𝑥subscript𝑅superscript𝑠𝑎𝑥subscript𝑅superscript𝑠𝑎𝑥subscript𝑅superscript𝑠𝑎\left(\begin{array}[]{cccccc}\times&-xR_{s^{\prime},a}\cdots-xR_{s^{\prime},a}&\cdots&1-xR_{s^{\prime},a}\cdots 1-xR_{s^{\prime},a}&\cdots&-xR_{s^{\prime},a}\cdots-xR_{s^{\prime},a}\end{array}\right)( start_ARRAY start_ROW start_CELL × end_CELL start_CELL - italic_x italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a end_POSTSUBSCRIPT ⋯ - italic_x italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL 1 - italic_x italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a end_POSTSUBSCRIPT ⋯ 1 - italic_x italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL - italic_x italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a end_POSTSUBSCRIPT ⋯ - italic_x italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY )

We now add the very first row (that equals (1,1,,1)111(1,1,\ldots,1)( 1 , 1 , … , 1 )) xRs,a𝑥subscript𝑅superscript𝑠𝑎xR_{s^{\prime},a}italic_x italic_R start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a end_POSTSUBSCRIPT times to this row and obtain

𝐰s=(×|00||11||00)subscript𝐰superscript𝑠|00||11||00{\bf w}_{s^{\prime}}=\left(\begin{array}[]{ccccccccccc}\times&|&0\cdots 0&|&\cdots&|&1\cdots 1&|&\cdots&|&0\cdots 0\end{array}\right)bold_w start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL × end_CELL start_CELL | end_CELL start_CELL 0 ⋯ 0 end_CELL start_CELL | end_CELL start_CELL ⋯ end_CELL start_CELL | end_CELL start_CELL 1 ⋯ 1 end_CELL start_CELL | end_CELL start_CELL ⋯ end_CELL start_CELL | end_CELL start_CELL 0 ⋯ 0 end_CELL end_ROW end_ARRAY )

Next we fix ssuperscript𝑠s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and rsuperscript𝑟r^{\prime}italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT such that rCssuperscript𝑟subscript𝐶superscript𝑠r^{\prime}\in C_{s^{\prime}}italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT and substitute the first row of

(×(Brj)jC1(𝐈δrjBrj)jCs(Brj)jCm)subscriptsubscript𝐵superscript𝑟𝑗𝑗subscript𝐶1subscript𝐈subscript𝛿superscript𝑟𝑗subscript𝐵superscript𝑟𝑗𝑗subscript𝐶superscript𝑠subscriptsubscript𝐵superscript𝑟𝑗𝑗subscript𝐶𝑚missing-subexpression\left(\begin{array}[]{cccccc}\times&(-B_{r^{\prime}j})_{j\in C_{1}}&\cdots&({\bf I}\cdot\delta_{r^{\prime}j}-B_{r^{\prime}j})_{j\in C_{s^{\prime}}}&\cdots(-B_{r^{\prime}j})_{j\in C_{m}}\end{array}\right)( start_ARRAY start_ROW start_CELL × end_CELL start_CELL ( - italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ∈ italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL ( bold_I ⋅ italic_δ start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j end_POSTSUBSCRIPT - italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ∈ italic_C start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL ⋯ ( - italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ∈ italic_C start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW end_ARRAY )

by the sum of the rows iBr𝑖subscript𝐵superscript𝑟i\in B_{r^{\prime}}italic_i ∈ italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Since for every s′′superscript𝑠′′s^{\prime\prime}italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT it holds that Qr,aj1=Qr,aj2subscript𝑄superscript𝑟subscript𝑎subscript𝑗1subscript𝑄superscript𝑟subscript𝑎subscript𝑗2Q_{r^{\prime},a_{j_{1}}}=Q_{r^{\prime},a_{j_{2}}}italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT for all j1,j2C¯s′′subscript𝑗1subscript𝑗2subscript¯𝐶superscript𝑠′′j_{1},j_{2}\in\bar{C}_{s^{\prime\prime}}italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT this sum has the following form

(×(xQr,aj)jC¯1(δ¯rjxQr,aj)jC¯s(xQr,aj)jC¯m),subscript𝑥subscript𝑄superscript𝑟subscript𝑎𝑗𝑗subscript¯𝐶1subscriptsubscript¯𝛿superscript𝑟𝑗𝑥subscript𝑄superscript𝑟subscript𝑎𝑗𝑗subscript¯𝐶superscript𝑠subscript𝑥subscript𝑄superscript𝑟subscript𝑎𝑗𝑗subscript¯𝐶𝑚missing-subexpression\left(\begin{array}[]{cccccc}\times&(-xQ_{r^{\prime},a_{j}})_{j\in\bar{C}_{1}}&\cdots&(\bar{\delta}_{r^{\prime}j}-xQ_{r^{\prime},a_{j}})_{j\in\bar{C}_{s^{\prime}}}&\cdots(-xQ_{r^{\prime},a_{j}})_{j\in\bar{C}_{m}}\end{array}\right),( start_ARRAY start_ROW start_CELL × end_CELL start_CELL ( - italic_x italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL ( over¯ start_ARG italic_δ end_ARG start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j end_POSTSUBSCRIPT - italic_x italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL ⋯ ( - italic_x italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW end_ARRAY ) ,

where δ¯rj=1subscript¯𝛿superscript𝑟𝑗1\bar{\delta}_{r^{\prime}j}=1over¯ start_ARG italic_δ end_ARG start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j end_POSTSUBSCRIPT = 1 if and only if jBr𝑗subscript𝐵superscript𝑟j\in B_{r^{\prime}}italic_j ∈ italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT and =0absent0=0= 0 otherwise. This means, for every s′′ssuperscript𝑠′′superscript𝑠s^{\prime\prime}\neq s^{\prime}italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ≠ italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT the entries (xQr,aj)jC¯s′′subscript𝑥subscript𝑄superscript𝑟subscript𝑎𝑗𝑗subscript¯𝐶superscript𝑠′′(-xQ_{r^{\prime},a_{j}})_{j\in\bar{C}_{s^{\prime\prime}}}( - italic_x italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ∈ over¯ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT are either all equal or if s′′=ssuperscript𝑠′′superscript𝑠s^{\prime\prime}=s^{\prime}italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT then we have to add 1 at proper positions. For every s′′superscript𝑠′′s^{\prime\prime}italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT we now add row 𝐰s′′subscript𝐰superscript𝑠′′{\bf w}_{s^{\prime\prime}}bold_w start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT xQr,aj𝑥subscript𝑄superscript𝑟subscript𝑎𝑗xQ_{r^{\prime},a_{j}}italic_x italic_Q start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT times. If s′′ssuperscript𝑠′′superscript𝑠s^{\prime\prime}\neq s^{\prime}italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ≠ italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT then we get a zero block (0,,0)00(0,\ldots,0)( 0 , … , 0 ). If s′′=ssuperscript𝑠′′superscript𝑠s^{\prime\prime}=s^{\prime}italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT we get a block of the form

(001100).001100\left(\begin{array}[]{ccccc}0\cdots 0&\cdots&1\cdots 1&\cdots&0\cdots 0\end{array}\right).( start_ARRAY start_ROW start_CELL 0 ⋯ 0 end_CELL start_CELL ⋯ end_CELL start_CELL 1 ⋯ 1 end_CELL start_CELL ⋯ end_CELL start_CELL 0 ⋯ 0 end_CELL end_ROW end_ARRAY ) .

This means that this row is replaced by

𝐰s,r=(×|00||00|00 11 00|00||00).subscript𝐰superscript𝑠superscript𝑟|00||00|0011 00|00||00{\bf w}_{s^{\prime},r^{\prime}}=\left(\begin{array}[]{ccccccccccccccc}\times&|&0\cdots 0&|&\cdots&|&0\cdots 0&|&0\cdots 0\ \cdots\ 1\cdots 1\ \cdots\ 0\cdots 0&|&0\cdots 0&|&\cdots&|&0\cdots 0\end{array}\right).bold_w start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL × end_CELL start_CELL | end_CELL start_CELL 0 ⋯ 0 end_CELL start_CELL | end_CELL start_CELL ⋯ end_CELL start_CELL | end_CELL start_CELL 0 ⋯ 0 end_CELL start_CELL | end_CELL start_CELL 0 ⋯ 0 ⋯ 1 ⋯ 1 ⋯ 0 ⋯ 0 end_CELL start_CELL | end_CELL start_CELL 0 ⋯ 0 end_CELL start_CELL | end_CELL start_CELL ⋯ end_CELL start_CELL | end_CELL start_CELL 0 ⋯ 0 end_CELL end_ROW end_ARRAY ) .

With help of these rows we can eliminate all further entries of 𝐌(x,𝐚)𝐌𝑥𝐚{\bf M}(x,{\bf a})bold_M ( italic_x , bold_a ) that come from 𝐆(x,𝐚)𝐆𝑥𝐚{\bf G}(x,{\bf a})bold_G ( italic_x , bold_a ). (Here we use the fact that each row of Brr′′subscript𝐵superscript𝑟superscript𝑟′′B_{r^{\prime}r^{\prime\prime}}italic_B start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is either zero or all entries are the same.) This means that we finally end up with a matrix of the form

𝐇=(111×H11H1m×Hm1Hmm),𝐇111subscript𝐻11subscript𝐻1𝑚missing-subexpressionsubscript𝐻𝑚1subscript𝐻𝑚𝑚{\bf H}=\left(\begin{array}[]{cccc}1&1\cdots&\cdots&\cdots 1\\ \times&H_{11}&\cdots&H_{1m}\\ \vdots&\vdots&&\vdots\\ \times&H_{m1}&\cdots&H_{mm}\end{array}\right),bold_H = ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 ⋯ end_CELL start_CELL ⋯ end_CELL start_CELL ⋯ 1 end_CELL end_ROW start_ROW start_CELL × end_CELL start_CELL italic_H start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_H start_POSTSUBSCRIPT 1 italic_m end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL × end_CELL start_CELL italic_H start_POSTSUBSCRIPT italic_m 1 end_POSTSUBSCRIPT end_CELL start_CELL ⋯ end_CELL start_CELL italic_H start_POSTSUBSCRIPT italic_m italic_m end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ,

where Hss′′=𝟎subscript𝐻superscript𝑠superscript𝑠′′0H_{s^{\prime}s^{\prime\prime}}={\bf 0}italic_H start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = bold_0 for ss′′superscript𝑠superscript𝑠′′s^{\prime}\neq s^{\prime\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT and Hsssubscript𝐻superscript𝑠superscript𝑠H_{s^{\prime}s^{\prime}}italic_H start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is of the form

Hss=(JKKK𝟎J𝟎𝟎𝟎𝟎𝟎J).subscript𝐻superscript𝑠superscript𝑠𝐽𝐾𝐾𝐾0𝐽00missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression000𝐽H_{s^{\prime}s^{\prime}}=\left(\begin{array}[]{ccccc}J&K&K&\cdots&K\\ {\bf 0}&J&{\bf 0}&\cdots&{\bf 0}\\ \vdots&&\ddots&&\vdots\\ \vdots&&&\ddots&\vdots\\ {\bf 0}&{\bf 0}&{\bf 0}&\ldots&J\end{array}\right).italic_H start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL italic_J end_CELL start_CELL italic_K end_CELL start_CELL italic_K end_CELL start_CELL ⋯ end_CELL start_CELL italic_K end_CELL end_ROW start_ROW start_CELL bold_0 end_CELL start_CELL italic_J end_CELL start_CELL bold_0 end_CELL start_CELL ⋯ end_CELL start_CELL bold_0 end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋱ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_0 end_CELL start_CELL bold_0 end_CELL start_CELL bold_0 end_CELL start_CELL … end_CELL start_CELL italic_J end_CELL end_ROW end_ARRAY ) .

with

J=(111101000001)andK=(111100000000).formulae-sequence𝐽11110100missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression0001and𝐾11110000missing-subexpressionmissing-subexpression0000J=\left(\begin{array}[]{ccccc}1&1&1&\cdots&1\\ 0&1&0&\cdots&0\\ \vdots&&\ddots&&\vdots\\ \vdots&&&\ddots&\vdots\\ 0&0&0&\ldots&1\end{array}\right)\quad\mbox{and}\quad K=\left(\begin{array}[]{ccccc}1&1&1&\cdots&1\\ 0&0&0&\ldots&0\\ \vdots&\vdots&\vdots&&\vdots\\ \vdots&\vdots&\vdots&&\vdots\\ 0&0&0&\ldots&0\end{array}\right).italic_J = ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 end_CELL start_CELL 1 end_CELL start_CELL ⋯ end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋱ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL … end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) and italic_K = ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 end_CELL start_CELL 1 end_CELL start_CELL ⋯ end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL … end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL … end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ) .

It is now an easy task to transform the matrix (Hss′′)1s,s′′msubscriptsubscript𝐻superscript𝑠superscript𝑠′′formulae-sequence1superscript𝑠superscript𝑠′′𝑚(H_{s^{\prime}s^{\prime\prime}})_{1\leq s^{\prime},s^{\prime\prime}\leq m}( italic_H start_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 1 ≤ italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_s start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ≤ italic_m end_POSTSUBSCRIPT (with help of row transforms) to the identity matrix. Furthermore we can transform the very first row (1,1,,1)111(1,1,\ldots,1)( 1 , 1 , … , 1 ) of 𝐇𝐇{\bf H}bold_H to (1,0,,0)100(1,0,\ldots,0)( 1 , 0 , … , 0 ) and end up with a matrix of the form

(100×10×01).1001missing-subexpression0missing-subexpression0missing-subexpression1\left(\begin{array}[]{cccc}1&0&\cdots&0\\ \times&1&&0\\ \vdots&&\ddots&\vdots\\ \times&0&&1\end{array}\right).( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL × end_CELL start_CELL 1 end_CELL start_CELL end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL × end_CELL start_CELL 0 end_CELL start_CELL end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) .

Obviously, this matrix has determinant 1111. Since the above row transforms do not change the value of the determinant we, thus, obtain det𝐌(x,𝐚)=1𝐌𝑥𝐚1\det{\bf M}(x,{\bf a})=1roman_det bold_M ( italic_x , bold_a ) = 1.

Acknowledgement. The authors want to thank Philippe Flajolet for several discussions on the topic of the paper and for many useful hints.

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